How does Arthur canonically travel after Polite Society, Valentine Style?

At the start of Chapter 2, the Uncle mission naturally feels like the first mission to do, since it's the only one in the Camp.

When Arthur gets recognized in Valentine by the man from Blackwater, he jumps on a random stranger's horse to pursue the man. So logically, he hasn't called his own horse with him to Valentine - he would've simply pursued with his own horse.

After the man is dealt with, I believe Arthur would return the horse to the stranger. This is because he wouldn't want to cause trouble in Valentine to keep the gang safe, and being recognized as a horse thief would not be ideal.

With this same rationale, Arthur wouldn't steal a horse in Valentine to head out of the town.

Also, he wouldn't hitch a lift from another traveller, that would raise suspicion when he gets off in the middle of nowhere (near the camp).

So - how does Arthur travel out of Valentine canonically?