I carry in the woods what I carry on the street; which is a .45 ACP. Used to carry a 9mm Hi Power, and it served me well in the woods too. When using .380 or 9mm on bunnies, make sure you always use hollow points. I found out the hard way that fmj's don't anchor even a squirrel in 9mm. Switch to a JHP, and now you have to be careful not to spoil all the meat.
Best choice for a small game handgun is a .38 Special using target wadcutters. Cuts a nice clean .357 caliber hole all the way through, putting them right down.
