If the way the rifle is kept in the stock is the same as the factory stocks this may help.
Remove the receiver from the stock and remove the trigger group from the receiver of your 10/22. Sit the receiver back into the stock and look from the trigger side opening. Is there a ridge in the very back, just under the receiver where it would protrude into the profile relief of the trigger group ie directly under the back of the receiver? If so, you can bed that part of the action to remove the slop.
Mix up some JB Weld and set it aside. You don't need a lot, about 3/4" length squeezed out. It's a little runny at first so by the time you've done the prep work, it would have thickened slightly as it starts setting.
Take a Dremel with a small engraving bit and rough up the area directly below the ridge. Wipe the area worked on with some acetone or isopropyl alcohol to degrease it.
Next, tape off the trigger guard area about 1/2"-3/4" below that ridge and over the lip of the trigger area and to the point where that ridge blends into the side of the stock. Tape around the grip area of the stock below the trigger area as well, just in case of drips or squeeze out from the JB Weld.
Apply a generous amount of paste wax or neutral shoe polish to the back inner/under/back of the receiver in the general area that will make contact with the worked on area of the stock.
Apply a generous amount of paste wax to the back end of the trigger assembly and then reattach it back to the receiver.
Hopefully the JB Weld would be slightly more firmer and not as runny by now.
Apply a liberal amount of the JB Weld under the ridge of the stock. Do NOT discard the remaining JB Weld and stick you used to apply it. Set it aside and use it as a gauge to see how it is curing.
Carefully reinstall the receiver back into the stock and lightly screw in the receiver screw. Verify the barrel is centered in the stock's channel and lightly snug down the receiver screw.
Set your alarm clock for 5hrs.
Set aside to cure almost vertically ie butt down/muzzle to the ceiling. Inspect it in about 45min, ensuring that the barrel is still centered. If all's good so far, go kill 4.5hrs.
Check the surplus JB Weld you set aside earlier. Is it about 90% cured ie if it is stiff but still slightly pliable, great. If it still soft or firm, let it cure for another 30-45mins.
Remove the receiver from the stock. You may need to use some wiggling to get to to break free of the JB Weld. This is when you hope you used enough paste wax on the receiver.
Remove the masking tape and with a sharp knife, clean off any JB Weld flashing in the area and inspect the bedded area. Did the JB Weld contour nicely around the trigger assembly? If all looks good, reattach the receiver and lightly tighten the receiver screw. Set the rifle aside to cure another 12hrs or so before torquing the screw to spec.
Your receiver should no longer be wobbling in the stock.
I'll post pic later.
