I was going to explore with my horse who's sitting at #1 on the most pictures leaderboard when I misclicked "go back to main explore page". I never got to make a move before I accidentally clicked off. Now I cannot explore with then him today which may knock him down on the LB. Do I just have to wait until tomorrow to explore with him then? Thank you.

Yep. Once you click out, that's your "ride" completed.

Darn. Thank you for answering!