It changes daily on the two horses you can ride in the stables. Once you have completed the lessons from the board in the stables, you can free ride tati or blaze. On the board you can see which horse is available for what and from random days you can go out to the cross country. Doesnt happen everyday and its random. Just a case of on the day seeing.

I've been playing the game for several days in a row, but I haven't discovered any new opportunities with the track. :(

Is this in-game days or real world days?

It's calendar days, and the games uses your computer's calendar, so you can just change the date to change what options are available
As an example, I just opened the game and had Tati in the flatwork arena and Blaze on the XC grounds, then changed the date to the 22nd and had it the other way around.

You need to exit the game, then change the date, then reopen it ^^