Handgun hunting is a totally different world than hunting with a rifle. I've done a fair amount of it and finally decided that a rifle kills so much quicker and is so much easier to shoot accurately than a handgun that I'm primarily a rifle hunter again.
All that being said I found that my my favorite after owning, shooting and hunting everything from 357 magnum all the way up to 500 S&W magnum that my 10mm Glock Model 20 is so incredibly easy to draw and shoot quickly and accurately and 14 rounds in the magazine gives me a warm fuzzy feeling that I can quickly deliver a rain of vicious projectiles that very few animals smaller than a Cape Buffalo would be able to face.
Shot placement is incredibly important with a handgun and you're shooting a gun that is WAAAAAY harder to shoot accurately and the heavier magnums will recoil so much that even when your adrenaline is running wide open you will have a LOT of trouble getting off killing rounds quickly. Enter the 10mm cartridge. It propels a medium (for caliber) weight bullet nearly as fast as a 41 mag but due to the Glock's low recoil and design you can dump a magazine pretty darn quickly if you have reasonable strength and experience.
I know that one shot one kill is paramount when hunting medium to large game and spray and pray is the mark of a rank amateur but if I can't take that one killing shot with a great certainty that a quick follow up shot won't be necessary then I opt for medium killing power (10mm fits that bill well) and the ability to deliver a lot more rounds quickly may make for a messy kill but at the point of a ----ing match for who's top of the food chain it becomes, at least for me, a final outcome of who wins with the least amount of damage then I'll reach for that Glock 10mm every time and won't worry much about a bloody carcass with little salvageable meat.
By the way. I used to live in south Florida and killed a boat load of hogs with my Glock Model 20 10mm and I was shocked at how efficiently it dispatched them but then my shots were usually 10-30 yards and quick follow up shots are easy even with a handgun at that range. The exception was of course the big boars and then nothing short of my 300 mag impressed them much unless I went for the classic under the ear shot which can be pretty tough to deliver because most shots when hunting hogs on foot are on moving targets. At least for me they were and one was never enough hogs because I had hungry neighbors to feed.
I know I've mixed hunting with self defense here but I've done both and the 10mm cartridge does it for both for me unless I'm hunting elk, moose, or bear and not carrying a rifle as well.
Once again... A 308 rifle with a 150-180gr bullet is a WAAAAAAAY better choice for hunting medium to large game and ANY handgun is only marginal even for self defense. If a bear or other big nasty critter gets ----ed enough at you to sort you out in a permanent fashion the handgun is probably not going to get deployed in time to do much good. Most large game can move in excess of 30mph and that doesn't give you much time to do any more than say oh crap!
