Well, what can you say, Im a Mosin User myself, tizz my daily companion.
I use an M-39 as guys here probably know, and before I ever had to touch a rifle, other than in maintainace, to start with , I tighten the action screws and take it out for a test fire.
Of the three pillars to accuracy, I knock off the first two BEFORE I do any sort of improving.
I have the skills to make accurate shots, no flinching, no closed eyes, check out the trigger and the point of impact and see what I might do to improve my stance, hold, to use the sling or not and shoot at various range's.
At that time I shoot from the available ammo I have, and try different production years and countries of origin, as my M-39 loves 70's Russian and 50's Czeck, and the results in accuracy are worthy of my bi-athalon high school days scores.
Lots of practice, with pellet guns to the Mosin its self are used to sharpen and perfect your shooting skills until they are no longer a factor, I pay more attention to the ammo and finding consistent ammo at that. The closer in consistency with FPS variations the more tighter and predictable the groupings the ammo will make.
If I cannot achieve excellent results with various ammo, a basic check over and action tightening, THEN I start to do the things I need to do to achieve better results with changes and additions....................., but a first go around with your rifle will tell you many things about it and at least establish a 'base line' to measure your improvements from, and to be sure they are not actually causing you to lose accuracy or makeing it a dangerous tool.
Still, only accurate rifles are interesting, and this is the time to get a Bargain Rifle with bargain priced ammo. NO "Cheap" by any means, a Mosin Nagant will shoot up to any shooters skill, the quality of ammo and how well its taken care of. Indeed, a tweek and a carefull tuning can get better results, if its not allready there, but be sure to see if it isnt allready for ya
My M-39s have only once needed work, and 3 shims under th action (I belive they must have been there before and fell out on a previous owner) and its a tack driver.
Mosin Nagants can be as accurate as any other rifle on Earth, too bad we couldnt ask Simo Hayha