The Mosin-Nagant tends to be an inherently accurate design as illustrated by the Finns. Problem is that most were poorly executed by the Russians as precision was not high on the priority list. Here's a brief look at what I like to do to mine!
The first thing you want to do is smooth the barrel channel using an appropriate sized dowel rod or socket from a ratchet wrench, wrapped in 320 or so sandpaper. When you are done, you should be able to lay a straight edge along the barrel channel.
Lightly oil the cork bedding on both sides.
Next, cut cork, lay it in, and lay the barreled action into the stock. Observe whether the barrel lays flat. It may. Now, tighten the front and rear screws to about 50in-lbs. The barrel will probably raise the muzzle quite a bit.
Bring lots of ammo and tools to the range. This takes a lot to get right.
Start placing cork material in the front and rear. You want the barrel to lay as flat as possible when the action screws are tightened.
On mine, I could not get the barrel to lay perfectly flat, as there is an as-yet undiscovered pressure point someplace in front of the front action screw. It raises just a little bit.
I used business card stock on the top handguard. Oil it well.
This is not necessarily a bad thing though. It provides an opportunity to add another known pressure point.
Move the front sling mounting point back!
And this is critical: You have made the barrel bed and handguard one with the barrel. You have a heavy barrel now, for lack of a better term. It's bad for harmonics when you attach anything to the barrel, so move the sling back, especially if you are like me and use the sling to shoot!