In D&D 5e, the daily rate of mounted overland travel is generally the same as on foot, because horses get tired and adventurers carry a lot of heavy equipment.
See the section Special Travel Pace in the DMG (p. 242–243). This section starts:
The rules on travel pace in the Player’s Handbook assume that a group of travelers adopts a pace that, over time, is unaffected by the individual members’ walking speeds. The difference between walking speeds can be significant during combat, but during an overland journey, the difference vanishes as travelers pause to catch their breath, the faster ones wait for the slower ones, and one traveler’s quickness is matched by another traveler’s endurance.
In the same section, the rule is:
- In 1 hour, you can move a number of miles equal to your speed divided by 10.
and then:
- For a fast pace, increase the rate of travel by one-third.
- For a slow pace, multiply the rate by two-thirds.
So an unencumbered horse with a speed of 60 could theoretically travel 6 miles in an hour at a normal pace. At a fast pace (a gallop), 8 miles per hour. That's "twice the usual distance for a fast pace", where "usual" means a creature with a speed of 30. This suggests that a riding horse with no rider, traveling alone, can cover 48 miles per day at a normal pace.
So the rule that "a mounted character can ride at a gallop for about an hour, covering twice the usual distance for a fast pace" seems to exist to allow for mounted travelers covering short distances quickly by using the mount's speed instead of "the usual pace", for up to an hour each day.
So, according to the rules, a traveler on a horse at a normal pace (3 miles per hour) will cover about 24 miles in an 8-hour day. If you make the horse gallop for an hour each day (fast pace for a horse being 8 miles per hour), that range increases to 29 miles. That's within the realm of what you would expect in real life, with a fast horse on good roads in fair weather.
Variant: Encumbrance
If you're using the encumbrance rule, a Riding Horse needs to be carrying less than 80 lbs of rider and equipment to get its full speed of 60. Loaded with between 80 and 160 lbs it has a speed of 50, and carrying between 160 and 480 lbs (its maximum carrying capacity) it has a speed of 30. A 200 lb adventurer in chainmail with a dungeoneering pack, longsword, and shield weighs in at about 325 lbs, so under this rule a horse's travel pace is usually the same as an unencumbered adventurer on foot.
3 miles per hour
Actions in Combat are not relevant to the travel pace rules. Rogues are faster movers in combat thanks to Cunning Action, but so are most characters as a character's walking speed reflects:
Every character and monster has a speed, which is the distance in feet that the character or monster can walk in 1 round. This number assumes short bursts of energetic movement in the midst of a life- threatening situation. (PHB 181)
[courtesy of MisterB in the comments]
Cunning Action and the Dash action do not affect travel pace. Compare this to a feature like the Totem Warrior Barbarian's Elk Aspect of the Beast which directly mentions travel pace:
Whether mounted or on foot, your travel pace is doubled, as is the travel pace of up to ten companions while they’re within 60 feet of you and you’re not incapacitated (see “Adventuring,” for rules on travel pace). The elk spirit helps you roam far and fast.
What about Longtrider?
Longstrider would increase your travel pace most likely. Travel pace of 3 miles per hour is derived from the common walking speed of 30 feet. According to the Dungeon Master's Guide:
In 1 hour, you can move a number of miles equal to your speed divided by 10.
Since longstrider increases your speed, your travel pace also improves during the duration (which by default is 1 hour).
Why can't I just Dash?
Combats rarely last more than a couple minutes and require a character to do a lot of things that would be very physically taxing should they be done for extended periods of time. In the rules for Chases from the Dungeon Master's Guide this is made apparent by this section on dashing (mentioned in HellSaint's answer):
During the chase, a participant can freely use the Dash action a number of times equal to 3 + its Constitution modifier. Each additional Dash action it takes during the chase requires the creature to succeed on a DC 10 Constitution check at the end of its turn or gain one level of exhaustion.
While this isn't directly related to travel pace it demonstrates that staying in combat mode for long periods of time (in order to Dash) could quickly result in exhaustion before you simply can't move further.
Best Answer
Only one 8 hour period. Any additional travel would require checks to avoid exhaustion
You are asking if a character/party can alternate 8 hours of travel with 8 hours of resting but, unforunately, a character can only benefit from a long rest once per 24 hours:
This means that the second "chunk" of travel would incur a check to resist suffering levels of exhaustion every hour because, even if the party pulled off the side of the road to "rest", they wouldn't receive the benefit of a long rest:
Put another way, you can travel as many hours in a day as you want. You get one long rest per day and 8 hours of travel per long rest "for free." Any travel beyond that requires you to make a check each hour to resist exhaustion, regardless of how you break up the travel.