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.
According to Dungeon Master's Guide p.68, conveniently available on the Wizards of the Coast's website as a PDF:
The Far Realm is outside the known multiverse. In fact, it might be an entirely separate universe with its own physical and magical laws. Where stray energies from the Far Realm leak onto another plane, matter is warped into alien shapes that defy understandable geometry and biology. Aberrations such as mind flayers and beholders are either from this plane or shaped by its strange influence.
The entities that abide in the Far Realm itself are too alien for a normal mind to accept without strain. Titanic creatures swim through nothingness there, and unspeakable things whisper awful truths to those who dare listen. For mortals, knowledge of the Far Realm is a struggle of the mind to overcome the boundaries of matter, space, and sanity. Some warlocks embrace this struggle by forming pacts with entities there. Anyone who has seen the Far Realm mutters about eyes, tentacles, and horror.
The Far Realm has no well-known portals, or at least none that are still viable. Ancient elves once opened a vast portal to the Far Realm within a mountain called Firestorm Peak, but their civilization imploded in bloody terror and the portal’s location—even its home world—is long forgotten. Lost portals might still exist, marked by an alien magic that mutates the area around them.
Player's Handbook, p.302, summarizes the same information.
Seeing a creature from the Far Realm can risk an individual's sanity (DMG p. 265).
Githzerai philosophers are aware of the existence of the Far Realm, and it may be connected to the aboleths somehow (MM p. 14).
Some users of wild magic are so afflicted due to contact with the Far Realm. (PHB p. 103).
Some warlocks use the ancient knowledge of beings of the Far Realm. Warlocks of the Great Old One may worship such unfathomable beings. (PHB p. 105).
Powerful, world-shattering beings known as the Elder Evils are speculated to be creatures of the Far Realm (Mordenkainen's Tome of Foes p. 234).
Far Realm lore from earlier editions of the game can also be used in your campaign. However, much about this plane is intentionally left undefined in order to set a mysterious and unsettling atmosphere, so lacking much rigidly defined lore may be to your advantage. You may find it useful to read the works of H.P. Lovecraft for inspiration.
Best Answer
Location of the Vast Gate
Far Realm was initially created by Bruce Cordell for the AD&D 2e module The Gates of Firestorm Peak. This module is dated 1996, so you cannot find any lore from, say, Planescape series of products. In the module, we learn that some long time ago, a so called "Vast Gate" was opened by Elder Elves and then had to be sealed, yet the sealing was imperfect and was re-opening periodically under certain conditions and the players try to close it completely.
The setting is described to be located near the Firestorm Peak, in "the Shirelands, at the southern foothills of the Mountains of Frost". Such a location is not defined in any of the official campaign settings of D&D. Cordell penned a number of different products at the time and these products occasionally refer to the same locations or people (forming something like a Cordell-verse), but each of these products were meant to be easily integrable to any campaign setting, so we really don't know when and where the Vast Gate was first opened within the D&D multiverse.
On the other hand, a number of people have compared some of Cordell's later work, which takes place in Greyhawk (3e's default setting), with the Cordell-verse generic products and concluded that there are both for and against arguments for integrating the Vast Gate into the history of Oerth. An example proposal that discusses the available data can be found on a 2010 post by user ripvanwormer on planewalker.com. Taking into account the date when the elves first came to Toril, he proposes that the Vast Gate could have played a role in that migration and hence considers sometime in the middle of -30,000 to -18,000 DR to be reasonable time for the accidental first contact of the Elder Elves with the Far Realm.
After the Closing of the Vast Gate
The Vast Gate is revisited in Cordell's article "Enter the Far Realm" published in Dragon magazine #330 (2005). At this point it is described to be no longer open. Yet occasional contacts with the Far Realm are said to happen at random locations, causing so-called "cerebratic blots" to appear in the Prime Material Plane and allowing travel. Yet again, no mention of any of the official D&D campaign settings is made, except for Eberron.
Likewise, if we look at the discussion in the 3e Manual of Planes:
The 4e Manual of Planes does not contradict with this description, mentioning that it only touches rarely the material world and that it is not normally a place one can visit.
Novels - Living Gate
This lore is somewhat contradicted by the Forgotten Realms short story Watchers at the Living Gate by Paul Park published in the D&D anthology Untold Adventures. It describes the so-called Living Gate, a tiny portal located in the Feydark, beneath the ruined eladrin metropolis of Cendriane. Cendriane was mentioned in the 4e Manual of Planes, but to the extent that I could track, the Living Gate was not mentioned anywhere beyond the short story.
Eberron Specific
Finally, if you are playing in the Eberron campaign setting, the Far Realm is incorporated as the Realm of Xoriat in Cordell's above mentioned Dragon article. Yet, please note that Eberron exists in its own micro-cosmology, so it is up to the DM how to interpret the Far Realm - Xoriat correspondence. [There has been a motto that everything that is in D&D is in Eberron but not possibly in a form you recognize.]