Most just use rifle loads for hunting when using rifle caliber handguns.
I have loaded some 125gr SSTs in .308 as well as 125gr Nosler ballistic tips (think they may have been discontinued because I haven't seen any available in a while) and 130gr Barnes TTSX. All provide good accuracy in reduced .308 youth loads using H4895. The only game I have seen taken with an SST was a doe (first deer) one of my nephews shot last year with a factory loaded Hornady custom light 87gr .243. He was using a left handed Ruger American Compact All Weather with an 18" barrel that he had just received for Christmas. It was a bang flop at a paced off 115 yards. I was amazed at the trauma caused by that little bullet in a reduced recoil load. At 125gr I would imagine the .308SST would be plenty of bullet for medium game.