Cleaning solution in the bore, if it's not removed with a dry patch and the bore is still wet when you start shooting could cause stringing. The first shot in such a situation will be high and each subsequent one lower until by about shots 3 to 5 it will settle down and shot a decent group. The same thing will happen if you leave ANY oil in the barrel and try to shoot it out. In fact, leaving oil in the barrel and shooting before removing it can damage the barrel so I NEVER leave a coating of oil in my barrels; it's just too easy to forget to remove it. A product like Prolix or Microlon Gun Juice will protect a barrel from rust and will not cause stringing.
By the way, I doubt a oversized hinge pin will help at all with stringing. I also, when shooting from the bench don't rest the bag near the hinge pin but usually about halfway down the forend and have found that this doesn't cause stringing for me. If the stringing is related to heating of the barrel, then the first shots out shouldn't string since the barrel will not yet be hot. If barrel heating was the problem, then it's a fault of the barrel (poor heat treating and stress relieving). I've had and shot at least 20 or 30 Encore and Contender barrels and thus far have never found an accuracy issue from barrel heating.
I would agree with other posters who said that just one or two or three groups isn't enough to determine that a barrel is stringing shots.