Best accuracy in benchrest matches has always been attained by following these two primary rules:
1. Use a cartridge with minimum recoil. They move the rifle the least while the bullet's going down the barrel.
2. Use a rifle with a 2-ounce trigger that's heavy enough to lay on its rests such that only its trigger is lightly touched by the human shooting it. The more us folks hold onto the rifle with all our wiggling bodies (heartbeat, muscle control) inconsistantly from shot to shot trying to manage a much heavier trigger pull, the bigger the groups are.
So, to buy something off the shelf that comes close to what you want to do, get a heavy weight Savage in .223 Rem., set/change its trigger pull to as light as possible, then reload the best ammo you can. This is probably the best way without having to go to larger bores which may well be best for serious competition in the upper levels of stool shooting.