The date on the hotel register when Battling Butler signs it is four days earlier than the date when Alfred signs it directly under Battling Butler's signature a few hours later that same day (November 2nd vs November 6th).
Buster and his valet take the train into "Mountain Girl's" hometown, so that Buster can confess that he's not the new champion; yet when circumstances require that his subterfuge continue, and he leaves to go to train for the fight vs. the Alabama Murderer, they drive away to the training camp in Buster's car.