I used the 55gr Trophy Bonded bullet out of a 220 Swift, but at 22-250 velocities. I had no problems killing a muley buck, an axis doe an two Blackbuck antelope. A friend uses the Hornady 60sp in his 220 swift (again, same thing)
I also used a 50gr Barnes XLC bullet in a little .221 Fireball carbine to shoot an 80# pig; made a 50 cal hole al the way through, dropped him right there.
I killed a 275# hog with a 75gr Swift Scirroco out of a 22/6mm wildcat. Your Handi may not have a fast enough twist for anything over 60gr, but thats OK. The Barnes 5o or 53 gr TSX will work just fine. Many folks like that Winchester 64gr PP bullet too.
Now, would I use one on a big money deer hunt? Probably not, but for run of the mill hunts, sure I would, and do. My current 22-250 is the Ackley Improved version but with a very fast 8" twist. I use the Hornady 68gr Match HP bullet and it shoots completely through coyotes and piles them up right there. The 22-250 should be used as a sniping round, avoiding running shots,or bad angles. It is not your "all-around" type of deer cartridge, but then, neither is the 243 or 6mm!