I agree about the rests. I have both both front and rear bought bags, and home made sand bags made out of blue jean legs. I use a combination of these. The surface you are shooting off of needs to have some give to it and it needs to be repeatable. The key to any accuracy shooting is doing every thing the same from shot to shot to shot. This includes how hard are you gripping the rifle or lack of gripping the rifle. Trigger pull has to be the same from shot to shot, the way the rifle sets in/on the "bags" has to be the same. Even how much pressure you have the rifle pushing into your shoulder will effect your shot to shot placement. Shooting a break open rifle offers challenges that a bolt rifles or even a falling block rifle does not. You have to place the rifle back into the bags exactly the same way after each shot and you have re-grip the rifle exactly the same way each time. It can be done, you just have to pay attention to the details. None of this has any thing to do with the actual rifle accuracy, it does have to do with consistency.
The other tips given as to cleaning and fouling the barrel apply too. Remember that every rifle is different - that is, they all like something different. Some like to be squeaky clean, some like being dirty or just a little fouled, some like a boat tailed bullet, some like a flat base bullet, different primers, different powders, etc. Shooting small groups is not easy, however some rifles are easier to do it with than others. Handis can shoot very well, but there are more variables involved in shooting them well than some other types of rifles.
Good Luck and Good Shooting
