Another option, me just thinking here and putting this out there. Perhaps leave the horse for dead earlier, go to hang the guy, when they gallop off he is hanging their struggling. His injured horse comes wandering up, comes to check him out, and he grabs the reins with his feet, reins being tied to the saddle horn. This would all happen in minutes, obviously, and you could have the already weak man struggle to keep his lame horse still. If his hands are bound, it only takes a few fingers really to hold onto the reins. The horse could be limping, wandering to follow the other horses but comes across him. I'm not sure if you want the guy to die or not, but said lame horse could bleed out and fall to the ground leaving the man to die rather gruesomely. Or he could struggle out of the noose and binding, and be stuck with a crippled horse.
I could see a horse that is confident with people coming up to one flailing in a tree, especially if it was his horse, but no horse would do it if it were flighty. I'm certain my horses would come up to me and nibble at my toes, although I certainly would have any sort of athletic ability to hand myself the reins, much less be asphyxiated and still be kicking.