The Handi-rifles can be a little problematic in the accuracy department, I know. I'd probably at least try some heavy loads and see if that seems to shrink groups much. (H110, 4227, 2400, whatever.) At least with 2400 you've got a range of power levels you can try. H110 doesn't like to go anything but full-throttle.
If you think there's any chance the rifle is over-bored, slug the barrel before you go much farther. You'll save a lot of effort with one simple step.
The other thing to try is to move away from the 240 gr. bullets. Those should work well in most barrels, but fast-twist rates like heavier (longer, actually) bullets and slow-twist barrels like ligher (shorter) ones, generally. I'd not be betting that's your problem, though.
As disappointing as it may be, I've seen some of the break-action single-shots that just did about 3" at 50...
Good luck!