I've had several rifles which did that vertical-string thing. The usual fix was much like Slamfire's comment. I generally only free-floated the forearm and inserted a shim at the front of the forearm. Only thick enough that it took about a five-pound pull to separate the barrel and forearm to allow insertion.
The most egregious case was my little Sako Forester .243 carbine with a Mannlicher stock. Five shots had each shot one inch above the previous. I found that the Mannlicher stock was a two-piece deal. I removed the barrel band and junked the front piece of wood. Trimmed the front of the remaining forearm. Installed the shim.
3/4 MOA, very reliably thereafter. Not uncommon to put three behind a dime.
No real reason for wild-eyed dispersion in a group from a sporter, if it's properly tweaked and tailored handloads are used. Sorry, but I pretty much take MOA or better for granted in my sporters--and mostly better.
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