Sounds odd that you go from 1.5" groups at 50 to 5 or 6 inch groups at 100. I haven't shot mine at 100 as yet, but your figures make me wonder about if some kind of destabilization is happening. Do you know the muzzle velocity of your load? If the bullet is passing through the sound barrier (app 1050 fps) this could be a problem, but if we are talkin gabout the Lee bullet it would surprise me, wadcutters are far more notorious for this, even when started below SOS they will destabilize after 50 yards. How are you sizing them, are you sizing at all? I find that both the bullets I'm using, shoot a tad tighter when I pan lube 'em and don't size. But I usually run 'em through my sizer because it's more convenient.
As to the power of the 38-55, even starting 250 grainers at 1300 fps you have the equvilent of a 44 Mag, 100 yards should be easy game range. The 38-55 had quite a reputation in times past as a cartridge good for deer and bear out to 150 yards, loaded to 375 Winchester velocities it should easily perform quite well to 200 yards. It also had quite a reputation on the target range for outstanding accuracy out to 300 yards.
It is no secret that the NEF 38-55 chamber/bore/groove dimensions leave something to be desired, and I've no question that these keep us from seeing the "Target Model" accuracy one might expect a rifle so named to possess. I'm quite pleased that mine will keep my freezer filled, and that is what I was searching for.
As a target rifle goes, I would never look to an NEF for that anyway. A target rifle in my opinion does not shoot 3 into less than 1", it shoots 10 into less than 1/2" in 2 minutes or less, 1/4" is even better. While I'm sure this won't hold with others here, my experience with the H&R/NEF rifle is about the same as that I've had with any run of the mill 94, 336, SSK and Mini 14 and a bunch of military bolt rifles I've ever shot; they shoot 1.5-4" at 100. They can be coaxed to do a bit better, but it'd take a lot of convincing to make me think that such can be done on demand, any time, everytime.
I read a lot about the fine groups some get, but most of the pictures I see of groups shot from the same gun over time run more to 1.5" than less than 1". I don't mind a 1.5" rifle, thats dang fine accuracy for a hunting gun or a plinker.
If you are shooting the Lee bullet, I don't think it will shoot well above 1400-1500 fps, if you are trying for higher velocity, try using a gas check design.
Don't know if any of that helps, hope it does in some way.