Why would you want to shoot small game while big game hunting. Seems a little redundant.
Bang..............there goes every deer in earshot.
I've used 6" Ruger 22 pistols (blued years ago, stainless MK II now), I've killed many grouse with neck shots and still got deer or elk during the hunt period. Grouse (or rabbits) make pretty good suppers until you get your big game.