My $0.02 worth - look for one made before the Great Patriotic War (WW-II) and check the muzzle to make sure the lands are not worn down (conscripts w/ steel cleaning rods are not the best thing for a rifle's health.) If it's counter-bored you should be ok - the counter-boring restores sharp lands at the "new" muzzle which is now .5" inside the barrel.
After that everything is up in the air. You can try other things like:
1. free floating the barrel (w/ strategically placed metal shims; sand down the barrel channel if you must)
2. bed the action - the Russians used oiled felt but some use bondo, or devcon (your rifle, do it your way [I just don't want to know about it])
3. do a trigger job - Mosin triggers are stupid-easy to lighten or smooth out w/ nothing more than brass/soda-can shims & Flitz and a polishing wheel chucked into a Dremel
4. put a pressure pad under the barrel - again, felt, cork, or what-have-you - if you're not happy w/ the free float
5. sort your ammo - experiment w/ .310, .311, & .312 and see which your rifle prefers. I recently sorted a box of milsurp and I had all 3 measurements within the first 5 rounds. (Yeah, your loved ones may worry a bit when they see you leaning over your boolits w/ a micrometer late into the evening, but it's the price you pay for smaller groups.)
6. Maybe replace the bolt's firing pin spring to decrease lock-time
Sometimes you'll come up with a dud rifle; sell it and get another.