depends on how many horses you have available
if you only have one, then the best option is to travel (over flat/ easy terrain) at the trot, app. 8 Mph/ 12 km/h. this can be done for hours on end so a (good) horse can travel over 100 miles/ 160 kilometres in one day
If the terrain is bad then the travel would be at the walk- some 4 Mph/ 6 km/hour. Even slower if the ground is broken, full of holes and you don't want to break a horses leg. Again this can be done for hours so 50 miles/ 80 kilometres can be covered in this way
examples: Bud and Temple Abernathy - Rode 4,500 miles from New York to San Francisco in 62 days in 1911. They were aged 11 and 7, and travelled alone.
If you have several horses available then the travel can be faster- changing horses to allow them some rest (not carrying the rider). this makes travel a little slower (having to lead the spare horse) but more riding hours are possible.
If you have several horses placed in relays (like the pony express or other courier type organisations- plenty of examples from China, Persia and Europe) then the only limiting factor is the stamina of the rider.
Xenophon (Cyropaedia) reported that King Cyrus the Great (599-530 BC) "experimented to find out how great a distance a horse could cover in a day when ridden hard, but so as not to break down, and then he erected post-stations at just such distances" - and in result a route of 2,750 km was travelled in seven days and nights by fast-relay couriers. It is unlikely a single man could do it- they would collapse on day 4 at maximum.
So even with an excellent set of horses placed in relays (and a rider with a steel bottom) I would expect that 1200 kilometres would be a maximum. Probably followed by a bottom transplant