I never used to carry a sidearm when hunting, until the feral hog population exploded, and the odds of running into groups of them became much higher. It was an excellent excuse to buy a 10mm Glock, which I had always been fascinated by, anyway. I like squirrel hunting with a bolt action .22, and a rimfire .22 is inadequate for hogs (Enabling 101, simple).
Also, my deer hunting is usually done from a blind, which I walk to in the dark. So I sling my rifle, which I consider to be almost useless up close and in darkness, because of the scope. I carry the 10mm holstered across my chest, for easy access, and I use a cap bill light, so I have both hands free to operate the G20. I've never used it, so far, but I like having it.
As far as needing a handgun to finish off wounded animals, I don't take shots that I'm not certain about, and I stay quiet till the animal has had time to bleed out. This has always worked for me, but anyone can make a mistake, I guess. If I did this on a regular basis, I would probably stop hunting.