Accuracy is a combination of good ammunition and a rifle in good condition.
As interesting as the .303 cartridge is from a historical prespective, its an evolutionary dead end.
Very accurate ammunition for the .303 does not exist to my knowledge. Hornady, Priv Partisan, Remington, Federal, corrosive military surplus (if you can find it) are your options for factory ammunition. The .303 went out of British service in the 1950s and other parts of the empire a little later so the military surplus market has long dried up.
No one makes modern match bullets for the .303 to my knowledge. Since it uses bullets with a .311 diameter instead of a .308 like the Win .308 or 30'06, there is not a widely used modern cartidge that shares commonality so ammo companies have long stopped making improvements to .311 bullets.
.303 Enfield Rifle was a battle rifle made often during wartime with loose tolerances to fight wars not to win precision matches. 3-4 MOA was acceptable accuracy when it was brand new. Years of corrosive ammunition use, poor storage only can make it worse.
I have a .303 No. 1 Mk III Enfield that was built in Lithgow, Austrailia in 1942, sporterized in the 1950s but never fired still covered in cosmoline when I received it as a gift several years ago. With various factory ammunition, I only got 4-6 MOA using Prvi Partisan, Hornady, and Remington.
With my best handloads, the best accuracy I could achieve was 3 MOA. I think that's the best you can hope for with such a old military surplus.