.338 Fed interests me, too, but I have to take recoil into consideration. I've shot a .338 Win in a M70 with a full-size scope, and it kicked with authority even tho it weighed 9 pounds or more. It was tolerable, but I wouldn't shoot it for fun. Being much lighter, a Handi chambered for the "little" .338 Fed pushing a 200-grain bullet is gonna let you know when you touch it off.
While I'm daydreaming of perfect Handis, how about .250 Savage or .257 Roberts? I know Fred made a .257R, and it turned out well. It seems like the .25 caliber is a perfect balance of adequate bullet weight for deer and mild recoil, until you get to the bigger .25s. .25-35 might be worth reviving in the Handi, having a rim for good extraction.