Bounty Hunter takes longer than the rest of the roles, but you still unlock the saddle pretty quickly.
One other feature of the game is that you can't switch stirrups between saddles, and you really want better stirrups when you unlock them, so wait until you get access to the saddle you want. I'm lvl 170 and have the reverse dapple silver Nokota (I think its called that - the one with the black face) with the bounty hunter saddle and hooded stirrups. I feel no need for further upgrades.