Here's my experience, for whatever it's worth.
I have a Remington 700 Stainless Synthetic Mountain Rifle in .30-06. It has the much-maligned "Tupperware"-type stock. These days I can put the first three shots into about an inch or less... at 200 metres. My secret is in the careful bedding (along with careful handloading).
What I've done is bedded the action in the conventional way. Those "Tupperware" stocks are notoriously difficult to bed because normal bedding compound won't stick to them. What I did was rough the hell out of the bedding area with a rasp and Dremel and then use an unorthodox bedding compound -- the heavy-duty variant of automotive Bondo called BondoGlass. This stuff has millions of tiny fiberglass particles embedded in it, and it is nasty to work with, but it dries REALLY quickly, so your bedding job will be complete within minutes. It worked well on this particular job.
I've also found that because of the ultra-skinny .30-06 barrel, the barrel whip on my Mountain Rifle needed to be dampened if I hoped to have any kind of consistent accuracy. The solution is to bed in a patch under the barrel about an inch behind the forend. Once the material was dry, I covered the bedding pad with a layer or two of black electrician's tape, so the barrel was not resting on something hard. The result was a tiny bit of pressure on the barrel at that point, with some "give" because the tape is slightly springy when compressed.
Man, does this system work!