Powerful devils like Asmodeus have never been mere 'pit fiends'. Indeed, none of the Archdevils nor their Demon Lord counterparts could possibly be mapped to anything in the 5e MM.
If you absolutely must have statistics, you'll have to create them yourself at this point.
We aren't completely in the dark on how to do it. Sword Coast Adventurer's Guide offers some insight into Asmodeus' character under his description as a deity. The Rise of Tiamat presents stats for an avatar of Tiamat, and Out of the Abyss has stats for several very different Demon Lords. Asmodeus' avatar (remember Asmodeus is a Forgotten Realms deity currently) should easily be on par with Tiamat's avatar and make even Demogorgon and Orcus look weak by comparison. Don't be afraid to throw out all the stops.
If you still need ideas on what powers Asmodeus should have, then I'd recommend looking up in entry in the 3.5e Book of Vile Darkness. It cannot be mapped directly to 5e, of course, but reading through it should give you an idea of who and what Asmodeus is.
As you said you're all right with consulting the lore of previous editions for an answer, I'd suggest taking a look at Planes of Law from Planescape, if you can get hold of it. (Not just for this particular question, but because it has an excellent presentation of Baator as a whole.) Fires of Dis would also be a great reference for Avernus and Dis in particular. The 3e Manual of the Planes also has a good overview of Baator that could help flesh out your planned adventure.
Anyway, on your particular question. According to a map provided in Planes of Law, as well as the individual descriptions in separate layers, there are the following inter-layer paths between layers (any without descriptions are referenced only on the map):
Further, there are direct portals from the Outlands to the city of Grenpoli in Maladomini, as well as from Sigil to both Grenpoli and Malagard. And of course, the main portal from Ribcage in the Outlands to Darkspine in Avernus. From Planes of Law: Baator, page 5, a general description of the state of portals out of Baator:
Most of the known gates [out of Baator] are two-way and are extraordinarily well guarded, preventing both entrance and egress. Anyone hoping to escape via these gates had better think twice, because the baatezu won't when they catch the poor sod. Most of these portals are build inside the fortresses of the major fiends of the layers. No one knows if the fortresses sprang up because of the portals, or if the portals just happened to appear in the fortresses after construction. 'Course, no one's really asking.
And from page 4, a description of the gate between Ribcage and Darkspine; this is from the perspective of entering Baator, but there's still some relevant portions:
The best route in is via a gate located in Ribcage, that "quaint little village" on the edge of the Outlands. The Cursed Gate, as it's called, is in a heavily fortified part of town; in fact, it's accessible only through the citadel of Lord Paracs, the ruler of Ribcage.
...
...the lord of Ribcage and the baatezu guards on this side of the gate have come to a tidy little agreement, one that profits both sides and doesn't bend the laws too far. Basically, the two parties bob travelers through the gate for whatever they can get...
Also, just as a note that's not especially relevant for you, earlier works suggested that the River Styx only connected to Avernus (as it connects to the first layer of all the Lower Planes) and Stygia. Of course, this isn't binding on later lore by any means, and that passage in the 5e DMG definitely does suggest otherwise as of 5e.
Best Answer
Welcome to Hell!
Actual lore and citations below
I want to preface that everything here is what I learned from reading Mordenkainen's Tome of Foes (MToF) and the Monster Manual (MM). I definitely recommend it as the story of the Blood War is absolutely fascinating plus you get to learn about how it could be extremely good or extremely bad for you if a True Neutral archmage like Mordenkainen just shows up one day. Everything here is what's canon for 5e, I can't speak to earlier editions.
tl;dr Every soul starts as a Lemure which is the only type of devil that can die in the Nine Hells and reform. Every other soul becomes a ranked devil of some sort, and with that increased power carries the threat of utter annihilation should they die in the Nine Hells.
Chad pulls out a scale with a feather on one end and unceremoniously drops you on the other.
You say, "Yay Mammon".
Your form is ripped to pieces and reformed into an Imp.
After several hundred years of various stalkings and the like, you are promoted through the ranks from Imp all the way to Bone Devil as you continue to excel at stalking Julia. Following this promotion, you almost immediately find Julia. Almost like she wanted you to find her.
You do that.
Two days later, Chad receives a letter via courier informing him that you have reported his violation of infernal code 66.33.311.8, subsection 122-5, paragraph 88.333, sentence Jeremy Bearimy, word 5. Almost immediately thereafter comes a form that has been stamped in triplicate which are the forms to have Chad carried away because you are technically correct, the best kind of correct.
In the next minute, you receive your promotion paperwork and key to your new office. Right after, an Imp from HR comes in to explain things.
You end up rocking this position by shrewdly selling junk souls on the bonds market and bailing out right before they crash screwing over both Levistus and Mephistopheles.
You end up being promoted to an Amnizu serving Mammon directly. Due to your absurdly broad knowledge of the infernal laws and contracts, you become aware that Mammon cannot legally ask for aid if a black cat stands atop a ladder within 400 feet of a the Chili's on 3rd Street on the blood moon while it rains.
So you orchestrate an insanely absurd series of events under his nose to cause this series of events to happen exactly when he's choking on a croissant.
And you merely respond, feigning sad resignation:
He dies.
Asmodeus, impressed by your scheming promotes you to Archdevil of that layer.
Two months later, you slip in the tub and die. Your soul destroyed. Maybe next time around you don't run that layer like Mammon and actually spend some money on repairs and safety features. The lack of handrails is certainly ominous, but definitely an OSHA violation.
Because it cannot be definitively proven that you slipped on ice that was put there by Levistus as revenge from your junk soul scheme, he is cleared of all charges.
From MToF, on page 17 there's an entire section on devil ranks and how creatures rise and fall. There's 13 ranks in all, so let's start at the bottom:
Unranked Devils
Lemures Sitting at the bottom, Lemures are created from the souls of evil mortals who more or less ooze out of the River Styx. Even in the Nine Hells, it's hard to kill a Lemure as it has the Hellish Rejuvenation feature which allows it reform after being killed. This is a big deal as it allows Lemures to regularly serve and die as shock troops within the Blood War without the Nine Hells losing too many resources.
In general, the Lemures are a commodity to be exchanged and the manner in which they are exchanges is determined by the Archdevil Mammon, whose forces patrol the River Styx to harvest the Lemures. Mammon then contracts the sale of the Lemures to other Archdevils per the terms of whatever other contracts apply.
Nupperibos Although more powerful than Lemures, these creatures are created from souls that are deemed absolutely worthless because their damnation stemmed from sloth. Thus, even if killed, the Nine Hells does not feel as though they've lost anything.
Lesser Devils
First of all, past this point all devils described fall under the authority of an Archdevil. Meaning, that if an Imp (Rank 2) that serves under Mammon is given an order by a Bone Devil (Rank 7) that serves Levistus, the order will have no weight and be unenforceable. However, were Asmodeus (Rank 13) to give any devil an order, they'd be bound to follow it because Asmodeus commands all devils.
It should be noted that the promotion from a Lemure, presumably includes an oath of loyalty to the Archdevil that is enabling the promotion. Per the MM:
Ranks 2-7 These ranks constitute Imps (2), Spined Devils (3), Bearded Devils and Merregons (4), Barbed Devils (5), Chain Devils (6), and Bone Devils (7).
Per the MM, promotion to any Lesser Devil requires the power of a Greater Devil or an Archdevil. Apparently there's a painful transformation to move the memories from the form of a Lemure into their new form. Promotions to Imps are typically performed based on need, while higher promotions are typically based upon merit. Per MToF, merit based promotions stem from the devils' performance in the Blood War under the scoring category of Glory. Failure and disobedience as a lesser devil can entail punishment via demotion back down to a Lemure, which causes the loss of all memories.
Greater Devils
Greater devils are created from lesser devils and can only be made by Archdevils. It should be noted that this promotion carries substantial risk, because should demotion be necessary down the line the demotion will not remove the Greater Devils memories. Therefore, it may seek revenge.
Ranks 8-11 These ranks consist of Horned Devils and Orthons (8), Erinyes (9), Ice Devils and Narzugons (10), and Aminizus and Pit Fiends (11).
Presumably, promotion to these ranks stems not just from merit as described in MToF under Glory, but likely also include the Souls and Treachery criteria. Devils are expected to harvest souls and the higher the value, the better. An example provided for Souls includes a devil that manages to make a powerful warrior pledge their soul and form as an Ice Devil within the Nine Hells; obviously this soul is of much greater value than any random evil commoner. Thus, the devil that secures it has greater standing an is more likely to be promoted.
Alternately, there's the Treachery option, whereby a devil learns the complex rules of the Nine Hells and the applicable contingencies and causes a situation whereby there's a vacancy above them and they are defaulted. This sort of a trickery is wholly accepted and part of the game. This doesn't necessarily mean that the devil directly kills their superior, thereby destroying their soul (this is likely forbidden by infernal law), but rather they may cause their failure and thus mandate their demotion. Indirectly killing is probably completely permissible.
Archdevil
Not surprisingly, only Asmodeus can promote a Greater Devil to an Archdevil. Generally, this is a pretty rare occurrence and likely only happens via the Treachery option.
Archdevils rule a layer of the Nine Hells and have a full chain of command beneath them. There's no specific lore I could find regarding how this promotion process occurs, presumably it is the same as that for a Greater Devil, however, an Archdevil's form is wholly unique to their personality.
Once promoted to this status, the soul must learn how to be paranoid very quickly because at this point every devil under their command and outside their command is looking for a promotion.