360 is right!!

Low on dough.
Ken44, would you rec a .380 over the .22?
I don't know the size of animals your talking about(varies I assume). But with most any handgun load, esp a smaller one, a brain or spine shot will most likely be the only one that will dispatch the animal almost instantly. You may not even penetrate the skull 100% of the time if you do hit it with a .22lr. Through, or right behind the ear is where the old timers around here placed it when Deer hunting with a .22LR.
I really wouldn't suggest any pistol for that, unless you can be 100% accurate all the time. I don't think your the type that would want to take a chance of causing more suffering, and have to watch it.
About 10 years ago I saw an LEO pulled over looking over a wounded Deer.
I got out to see if he needed any help. He pulled out his Glock 17 and fired at least 6 rounds... he shot its jaw off and everywhere else around the brain but the right spot. He fired a few more and finally killed it, it wasn't a pretty site. I was about to ask him to let me try.
With a .22 rifle you can acually put the barrel right were it needs to be if the animal is seriously injured. You don't want to get as close as you would have to with a pistol to do that. And injured animal could seriously hurt you.
Sorry to ramble on a bit, but don't want you to do something that could upset you. Think about it, and if you have to go with a pistol, go with a larger caliber. .22lr or any will work, but placement will have to be exact with any.