In the past I have worked on several of these rifles in various cals. In all cases it was because they shot patterns not groups and in all cases the cure was the same. The rifles were bedded so that the recoil lug was bottoming in the stock and this was bowing the action when the screws were tight. The cure was to eliminate the contact between the bottom of the recoil lug and the stock.
Here is my .30/30 version of the 340.
Make: Sears Roebuck & Co.
Model: 101.53521
Action: Turn bolt
Stock: Hardwood walnut stained
Trigger: Factory
Weight: 7 lb. 3 oz. w/o magazine
Sights: Front, integral ramp and gold bead. Middle, U notch with step elevator. Scope, Weaver K4 with thin crosswires in Weaver side mount.
Accessories: Sling swivel studs. Extra magazine.
Modifications: Rifle rebeded in stock to improve accuracy.
Comments: This rifle is the same as the Savage Model 340.
Cast Boolit Info: Barrel slugged on 8/21/2009 and the groove diameter is .3087”
NOE 311407Mod, 50/50+2% tin, dip lubed in 45/45/10, H335-25.0G, 0.8G Dacron, SD=.558”, Crimp just removed flare, Chrono 1729 FPS. ½ to ¾” @ 25 Yds. depending on my state when shot.