I would carry and hunt with any of my handguns. However, their suitability for either scenario would shift depending on which one I'm talking about.
Not counting my wife's PPK/S (*sigh*...I'm still trying to talk her into giving it back), I have 4 handguns:
Automag II (.22WMR)
Colt 1991A1 (.45acp)
Beretta 92FS (9mm)
Colt Peacemaker (.45 Colt)
I routinely carry my 1991A1 and would certainly use it for hunting. I wouldn't have any problems hunting small game and varmint with it at reasonable ranges.
I have hunted small game and varmint with my Automag II, and it's got a respectable accurate range. For a carry weapon, it's mighty slim and easy on the weight. Not what I'd prefer simply based on the caliber, but it would get the job done.
I've carried my Beretta and it's plenty accurate enough to hunt with. Larger capacity than either of my above examples, but a wider frame than either.
I haven't carried my newest baby, the Peacemaker, yet. I didn't get it with a mind to carry it concealed, but I suppose I could. It's proving to be plenty accurate enough for me to use for small game and varmint at reasonable ranges, but I'd have to say it's not nearly as concealable as my semi-automatics. But it's REALLY COOL LOOKING and that has to be considered as well!
Overall, I'd say it's down to the Automag II and the 1991A1, and the 1991A1 wins out because of the self-defense edge.