I don'f often carry in AZ while hunting. If I happen to be carrying, it is a 9mm.
If I am in an area with a lot of "north bound" traffic then it is loaded with self-defense ammo. If I am in area with black bear, then it is loaded with Buffalo Bore.
Take whichever one you shoot the fastest and most accurately. Otherwise make sure the handgun has plenty of lube for when it gets shoved up your bum. No different that which rifle to take in grizzly country. You're better off taking your much maligned 6.5 CM that you can get a fast CNS shot each and every time rather than your manly 338 WM that you are scared of.
I'm carrying a HK in 45 acp. The slim version of the mod 23. Running plus p's with hard cast bullets for my back up. Eats the hard cast like a fat boy in a candy store. A 10mm would be better but I am set up for and familiar with the HK.
I think it matters entirely where you hunting. If there are bears or mountain lions in your area I'd want something with more punch.
I'm in the South East so I usually just carry my yard gun, a PX4 in 45 acp. If I were in a more dangerous area I'd want to carry a 454 Casull or 480 Ruger Super Redhawk. I used to own one in 454 Casull but sold it since it was more than my area needed & I had my eye on something else at the time.
Glock 20 when running in wyoming / montana / idaho / colorado / utah / new mexico / arizona ( north of i-10)
Glock 17 or 19 in texas & especially in arizona once you get south of I-10...with two extra mags.
Texas problem is meth labs & grow sites and occasionally hogs. Arizona is crews transporting drug loads north...same for texas on padre sea shore and southern border areas. None of the above are fun to run into and are usually in places most people won't go which mean places that dedicated hunters will go unfortunately. High % of the two legged issues have been in TX,AZ,NM. High % of the 4 legged issues have been in WY/MT.
Always have a light on it + some meprolight or trijicon sights.
Whatever you choose...practice with it...including drawing from holster. A lot. I almost never walk anywhere with my bow wrist loop on nowadays. Makes it impossible to get your pistol if you need it.