This is called making "Mexican match". I have done a lot of experimenting with doing this with 7.62 x 51.
Any soft point bullet will be more accurate than the military FMJ. Joyce Hornady once told me that their accptance standard for their FMJ was a group almost twice as big as for their 150 gr Spitzer soft point.
You mentioned accuracy. My testing suggests the following:
Pull the bullets and collect the powder.
Champfer the case mouths.
Run all the cases into a conventional neck sizing die with the decapping pin removed. (not a Lee collet.) This will give maximum neck tension and improve accuracy.
Add powder back and seat bullets.
Powder charges:
Reduce powder charge 1 grain to maintain the same pressure, which is usually in the order of 53,000 to 58,000 psi.
To improve accuracy, reduce the powder charge another grain. This ammo is great for plinking out to 600 yards.
To make real good target ammo, use a 155 gr match bullet and reduce the power charge 2 grains.
Once upon a time I bought a lot of 7.62 surplus for only $35/1000, so did a lot of work figuring out what to do with it. My results were all tested and approved for sale by the Explosives Branch.