Do yourself a favor and take that old sweet rifle over to Doc's in Carrollton 1 block east of 1-35E in the little white house across from the RR track and water plant and get him to look at it and tell you what he knows about it.
If it were mine, and I wish it were, 'cause I had one given to me from the late 1950's or early '60s after I first started hunting in 1966 with a cheap sporterized Springfield rifle from WW I, and the SAKO had been well used but had been sitting for an extended length of time, I'd make sure:
1. the bore is spotless and not gunked up with dust, old lube, lead deposits yadda yadda.
2. the crown is undamaged
3. the stock is not warped and putting pressure on the barrel
4. the trigger group is clean and not full of dust, burnt powder residue & accumulated dietrus.
5. make sure the scope mt's are tight and there are no issues with the scope.
My 243 SAKO shot late 60's era Rem/'chester factory ammo with my novice level skills at about 1 1/2- 2" groups....which was a significant improvement over the old Springfield. After I learned how to shoot correctly I got it to under the 1 1/2" groups size consistently, but whn I learned to reload even with the components from the late '60/early '70's I was able to shrink my group sizes to 5 shot that could be COVERED by a dime frequently at 100 yards. The only tuning I had done to that SAKO was to free float the barrel and have the trigger pull adjusted down to about 2lbs pull.
SAKO's are noted for their accuracy but every rifle has it own appetites. In current factory ammo I'd start with Federal and hunt with 90 or 100gr Fusion's, or Sierra Game King's or Nosler Accubond or even their plain solid base bullets - I am NOT a fan of the Nosler Ballistic Tip bullets on game, all under the Federal brand. Hornady Custom stuff is also pretty good in either the SST's or Spire's. In cheaper stuff I prefer Winchester over Remmy's & have had good experiences in Federal Blue Box stuff too.
Have fun and treat it like a treasure 'cause ya got a keeper.
Remember Aesops Fables of the "Scorpion & and the Frog's" crossing the river...
... we must hang together, or we will ALL hang seperately, Ben Franklin..
Those who fail to learn from History are doomed to repeat it,Santayana