Thirty-four years ago I hunted rabbits out on Long Island, NY with a Crosman .22 pump up pistol. The pistol was only supposed to be pumped up ten pumps max. I pumped it up to twenty (and blew the seals after a short time) but while it worked, it was powerful. Took eight rabbits with nine shots at distances of 10 to 25 yards.
A realistic concern in this day and age is legality. In a lot of states, hunting with and airgun is illegal. Check yours. You wouldn't want to run into a game warden carrying illegal dead bunnies.
Pete