I've hunted with a shotgun and handgun, but feel that a handgun was just extra weight and I rarely got an opportunity to use it. A few buddies have also lost handguns while carrying them in a holster.
Once, I saw a rabbit and went after it with the handgun in one hand and my shotgun in the other. A deer jumped up and ran off before I could use either weapon. That was the end of "Two-Gun" Picher.
The shotgun can dispatch about anything a .22 pistol can, if you really need to kill it. If you just want to see if you can do it, the pistol can be fun, but in my experience, will often wound more small critters due to poorly-placed shots than kill them.
Hitting a black and white target on the range is much easier than hitting a brown critter in the woods. Because it's necessary to focus on the sights instead of the target with a pistol, the target is often difficult to see, so shooters tend to focus on the animal instead and shots go wild.
I use a red-dot sight on my handgun and it's much more effective in the woods. Problem is, it's not handy with that big lump on top, so I only use it when hunting with the handgun-only.
Picher