carbineman, I'm curious if this barrel might be one of the slim profile older H&R Topper 30/30 barrels? I had one (more than one actually) over the years, and I found that as I fired more, the more the barrel heated up, and shots would get more random. They are actually quit a bit thinner than current barrels, and it doesn't take long before to heat up.
I learned to shoot 3-4 shots maximum and then put it down to cool while I shot something else. Accuracy was definitely good while it lasted though. And I once I learned to be patient with it and made sure the scope was solid (like you suggest), I never did worry about where the first shot was going to land.

BTW my favorite, and most accurate, load used a 165gr Speer PSP at a less than max velocity. Sorry I don't remember the recipe exactly but was with IMR 4064.
Cheers
Kerry