Does the stock have a pressure point at the end for the barrel? If so it sounds like the barrel is resting on a flat spot and moves horizontally after each shot. Float the barrel except the pressure point. Touch up the pressure point so the barrel fits in somewhat of a v channel to locate it, then after a few shots it will settle in.
Abstract exageration:
NO : _O_
YES : \O/
I had horizontal stringing with a Ruger M77 243. I now do this to all of my rifles with pressure points before I even shoot them.
You can also free float, but the rifle may like the pressure point. Also if you run out of material you may have to add some glass.
I use sand paper and a rod smaller than the barrel and lightly form a SLIGHT channel in the middle of the stock tip contact pad. When rifle is assembled you can hold the forearm and barrel and move to see if there is a channel or flat spot (We are not benchpressing 300 lbs when doing this). This is faint. I then punch the barrel into the stock to set it a little. A rubber mallet may work better and be easier on the hand, but I prefer the bottum of my clenched fist.
I have three rifles with pressure points that shoot sub moa. To me, proof's in the pudding.
