I wouldn't. The 10mm was designed as a defensive round, not for hunting. The bullets available are almost all made to expand on impact, rather than penetrate. A good hunting round needs to penetrate to the vitals more than it needs to expand and stop an attacker!
If you're hunting an area with Lions, or your hunting wild pigs, it'd probably be nice to carry on your hip. Or to keep handy for a coupe de gras (think i spelled that right?).
Thousands of deer have been killed with a .22 in the good old brain box. However, that doesn't mean an ethical hunter would use it. The foot lbs energy a 10mm generates are not enough to ensure a humane kill EVERY TIME, unless you're shooting the animal in the ear. And as we know, you don't want to limit yourself to an ear shot. So that's why you wouldn't want to use it for a primary weapon. For self defense in the woods, it's better than nothing, and a faster draw than a rifle. But as a primary choice - absolutely not. There are a good number of hunters who will never take a head shot becasue the off chance the animal will quickly glance away and you end up with a blown off jaw and a buck that starves to death.
Hunting is about respecting the animals we kill (I'm not saying that you don't - after all, you asked BEFORE you went hunting with your 10mm) and that means shooting them in the vitals, where you know they will die, and using a round with enough energy to kill them as quickly as possible.
The 10mm generates more energy with heavier bullets than the .357 magnum. Yes, it is primarily a self defense round, but make no mistake, this round has some serious power. There are bullets that work for deer. Hornady's XTP, in 180 grains at 1250 fps out of my Glock is NOT a self defense load. Nor is Cor-Bon's 180 bonded jacketed soft point.
If you are going to single out this round for non-inclusion in the handgun hunting rounds, then you must also eliminate the .357 and .41 magnums.
I'm not saying that the 10mm is the perfect hunting weapon, but you have to account for a lot more things using ANY handgun. A properly placed shot with a HUNTING (proper bullet/energy...don't go by commercial ballistics, except Cor-Bon) round from a 10mm auto will most certainly kill a whitetail cleanly.
I have hunted with mine, but have not had the opportunity to take a shot. Before anyone jumps my case, know that I am taking shots under 25 yards, and only clear shots to the heart/lung area.
Mister Venison makes the point that a lot of people don't know about. The 10mm is a very powerful round with a lot of potential, but the commercially avaliable rounds out there are paltry to say the least. Now smaller companies like Texas ammunition Company and Double Tap offer full power rounds, and of course you could always reload.
Double tap has a round that fires a Hornday 200grain XTP at 1250fps from a 4.6inch Glock 20 barrel, making 700 ft lbs. And of course retains its energy better than a 158 or 180 grain .357 bullet because of the weight. If you put in an after market barrel say in the Glock you even the playing field with the revolvers. You can get 5 or 6 inch barrels from Bar-sto or others, which would only increase velocity, accuracy and close the gap with the magnums even more.
As hog hunting gains popularity across much of the U.S. more and more hunters are asking where they should go to find the biggest hogs and trophies. I recently read an interesting article in "Boar Hunter Magazine", perhaps the preeminent hog hunting journal out there now. This very informative article was penned by Mr Jim Smith, a very accomplished hunter over a very wide range of areas and game. I am simply providing some information that should interest serious hog hunters among us. Mr Smith...