When Clay Hardin tells the driver to turn the coach around at the end, the long shot shows his horse tied to it. However, just before this shot, Bozic sees Hardin's 'empty' horse running free.
In the final fight between the Characters Hardin and Stuart, Hardin punches Stuart and he falls hitting his head on a large rock. The rock, being fake, moves.
Clay Hardin catches the stage riding his running horse. It's never explained how he was able to jump to stage and then lift his saddle off the horse. He wouldn't have been able to run his horse as he did in the clip (over a great distance) with an uncinched saddle.