Before shimming or any modifycations, try differnt ammo. Some old ammo is just made for a beat zone through a belt fed machine gun. The light ball surplus is usually accurate, and I find Russian 70's ball and 50's Czeck ball very accurate through my M-39 and other Mosins.
Some ammo is crap, but be sure its not loose action screws, flinching or any other changable condition.
Action shimming with thin metal, the Finns used brass, where the action screws pass through the stock, are to 'float' the action and barrel, as well, the left the hand guards a bit on the loose side,but being a too hot to hold in combat and too cold to touch in winter, the hand guard remains.
Triggers are tunable, and there is off the shelf stuff you can buy to make the trigger better, or follow the mods guys posting here do, either way, even a crappy trigger is useable with practise.
I preferr a longer barrel for shooting at running animals, as I swing through like shooting a shotgun at a passing goose. My M-39 has a couple more pounds than the average hunting rifle, but its rock stedy and the weight is very controlable, so accurate shooting is what results. The weight is very carrable for any grown man.
When Im shooting at a Caribou or such from 50-150 yards I hold at '6 oclock with a Caribou 's head being the dot on the "i" and dead on at 200 or so with my front sights level top edge being where the bullets gonna land. As well, over 250 yards, I use the rear sights slide, so I do not lose sight of the target with hold over.
The Mosin is NOT 'Cheap' in any manner, but quite a bargin you should take advantage of. The ammo can be bought in bulk and adding the simple step of first pouring boiling water down your barrel, scrub, boiling rinse, then your regular routine of cleaning and oiling. Boiling water drys itself, with the action left open while the rifle cools, and dissolves the salts that the priming spews down the barrel with every shot. Salts lie in the barrel attracting water, and hence rust, but water easily dissolves those salts and I never get a dark of rusty bore after routine cleaning and oiling.
If you have trouble with the bolt being hard to raise , AKA "Sticky bolt", then your chamber is dirty. Keep your chamber clean, but in a new refurrbed Mosin, the cosmoline thats in the chamber can be dryed and very hard to remove. It can also look shiney and new after a few shots, but the cosmo is something that MUST be removed........A 20 gauge brush and some solvent will srub up just great, and a chamber scrub is mandatory fopr any rifle, but "Sticky bolt" is your Mosin's way of saying "Clean me!!!"
You cant go wrong with a Mosin any more than you can with any other rifle.
The refurbs are all brought into Soviet specifycation, then slatherd in cosmo and varnish, then put away for WWIII. The Finn reworks ar noted for great triggers and sights, stocks and such. A Finnish Sniper, Simo Hayha, was the worlds highest scoreing Sniper, EVER....he use a Mosin for more than 1/2 those kills. The other top Snipers were Russian during WWII, as well as Olympic Gold being won by several nations useing the Msin is some configuration or another.
I hunt for a liveing and I use a Mosin Nagant M-39 Variation, it is a great tool.
