It is a very unpopular opinion, but I do much better with my springers (HW50s and HW77K) when I hold them tight to my shoulder. If I try the "artillery hold" my groups become 5-6 inches wide at 55 yards. If I hold them nice and tight, I can make both guns do 2 inch groups at 55 yards pretty consistently and even end up with sub 1 inch groups (again, 55 yards) every once in a while. Now, some will probably respond with "2 inches at 55 yards? that's horrible!" I would like to preemptively respond by reminding them this is a post about SPRINGERS, and springers are not capable of the same accuracy as PCPs. I, for one, am quite pleased when I can put all my shots within 2 inches of where I want them, on a regular basis, at 55 yards with a sub 12 foot pound springer.
Everything I'm saying about technique is, of course, after you have loctited screws, ensured your seals are good and your spring/gas ram isn't broken, and ruled out the scope as a possible problem.
With most physical skills in life, it seems repetition is what builds muscle memory and muscle memory is what leads to repeatability. What I am getting at is that the most important thing you can do is be very consistent with every shot, and spend a lot of time doing it. Springers are very sensitive to how they are held. I have found that if you can hold the gun the same way you were when you sighted it in, your chances of that pellet going where you want it to every time are dramatically increased. That is the hard part though, repeating every aspect of the shot perfectly.
When I shoot my springers, I am repetitive to the point of looking ridiculous. I pull it tight into my shoulder and if if it doesn't feel just just right, I will repeat that initial motion until the clothing isn't bunched up between me and the gun and it isn't too high or too low on my shoulder. I also make sure to hold the guns in the same place every time, judged by where the checkering/stippling patches are on the stock. I also don't wrap my thumb around the grip but leave the thumb rested against the back of the grip pointing up towards where the back of the action stops. There are a lot of nuances with springer shooting and learning how to shoot each one is an adventure. What is nice is how rewarding it can be to send a pellet downrange, instantly knowing where that pellet is going to go, depending on whether or not I was holding the gun the same way I was when I sighted in the scope.
Springers are like shooting three pointers in basketball. PCPs are like clicking a computer mouse. The skill required with springers has to be approached as a personal challenge and is so rewarding when it works out. PCPs are for when you just want to very precisely smack something with a pellet.