Hay BigBore: My regular hunting buddy and I usually go with shotguns and 22 pistols or shotguns and 357s. Rarely get the chance to bam one with the scatterguns as we're usually chasing them around heavy ground growth, stone walls or fences, brushpiles or brushlines where they can hide and you have to get close with a handgun.
With Snowhares, stalking slowly is about the best way if there is snow on the ground. Most bunnies, including the cottontails we hunt here try and avoid large open areas where they are subject to airborne predators and stick to places where they can get to cover quickly. If you want to foster the wabbit population, build some large brushpiles where they can find shelter or knock off some of the local bunny eaters so there is more for you. Sometimes they are very difficult to see in the snow but, look for the eyes and their tracks. If you see tracks you can expect to have some in the area. No tracks, no bunnies, no dinner.
357s with wadcutters or semi-wadcutters make for good bunny medicine and don't ruin a lot of meat. 22s are also real good. Good luck and have fun. And dinner, too. This is Mikey.