Awhile back, I bought an old late 80's era 10/22. That makes 5 in the stable..... But it had the metal TG and it was priced cheaper than a new one.....
Took it to the range and was I surprised/shocked to see that with AE 38 gr hp it was a ragged 10 shot hole at 20y.
Rifle was in OK shape (some would say poor), 75% blueing and some rust on the barrel, a bunch of paint missing on the reciever and trigger group, The stock was 'extensively' ding and scratched. I figgured that for a rifle this accurate it should look a bunch better. So I started in on it, I bought it for a project but this isn't the one I had in mind when I bought the rifle.
I tore it down and just my luck the local g-smith was just getting started with his annual blueing marathon. (He only hot blues once a year, it takes 3 weeks to get it all done, but sure saves time and effort on the set-up). I found a factory walnut (deluxe model) stock for a fair price.
By the time it was all said and done she is re-blued (just the barrel) re-painted (action and trigger group), and the stock is re-finished (danish oil).
Take it out to the range and she is 3.5" at 20 y.

Crappy, I just ruined a good gun!
I took it out again but this time I took the original stock with me. First thing I noticed was the walnut stock was sloppy as heck around the sides and back of the action. With the original Birch stock it is very tight, and I mean REALLY tight, take a rubber mallet to pop the action out of the stock tight. So I put the rifle back into its birch furniture and voila! 1/2" groups again. Swap out the stocks, and the walnut one is up over 3".....
So to see if bedding the walnut stock would acutually help I started layering masking tape in the action area of the stock to take the play out of the area the action sits in.
5 layers of masking tape on the sides and 7 at the rear of the action later........ 1/2" groups!!!!
In short, bedding the barrel has little effect on accuracy. Bedding the action has a TREMENDOUS effect on accuracy. Bed the sides of the action as well as the rear of the action where the stock has the shoulder that fits in the trigger group.
YMMV
Cheers!