I've no experience shooting jackrabbits, so I'll answer for eastern cottontails: subsonic .22lr. At long range, it often won't even alarm them if you miss.
Most all of my cottontail kills have been shot at long-ish range in a close cropped clover field. The last was about 35yds in the back of the head. My record is around 85yds with subsonics. He did an amazingly high backflip and was DRT.