The horse ridden by Kelly is the same one that Clint Walker rode most of the time in the television series, Cheyenne (1955). The horse, named Brandy, can be identified by the large white blaze and the white spot on its right hip.
Clint Walker did some of his own stunt work in the film. There's an excellent scene about 20 minutes in when Edd Byrnes has fallen from his horse, Walker wheels his horse around, gallops back, dismounts in reverse, hits the ground running into the foreground of the frame, takes his mark next to Byrnes and begins firing. All this is done in one shot and it's clear there was no stunt double.