990,
I was having the same result with #1 shot, and was using lightly coated barrel with bore butter. My local gun dealer, told me that the rep. for Thompson told him, not to use bore butter between shots, so I stopped. Has something to do with the faster twist rate on their in-line rifles. Now the only time I use bore butter is for storage after hunting season.
When I quit using bore butter between shots, and just load over a clean barrel, I pop one primer down the barrel for fouling, without a powder charge, and load. My first shot is very accurate this way.
I think what happens is that the bore butter gets down in the riflings so deep that it takes the first shot to clear it out, therefore all shots after that, are right on.
Might try a one primer foul with no bore butter, and see if that helps your first shot.
When I know I will be target shooting a lot during a week or so, after cleaning I will use rem oil in the barrel. It seems to be easier to get out with a dry patch, and after fouling with one primer, seems to keep first shot accurate.