I have been only shootin for about 3 years now (its kind of a new hobby to me), I go to ranges alot, and the shots i take are from 100,200,300 yards. I usually like to use my 30-06 (which i belive uses the same bullet as the 308), At 300 yars, Its hard as all heck trying to put ur sights right on the 15X10 in paper. I hate to sound like im a bad shooter but I think I am :-) , it takes ALOT AND ALOT AND ALOT of practise to shoot your rifle and make your bullet go where you want it to go esspecially at long ranges (in my case 300 yards).
I know you said you want to stay away from magnum calibers, but my friend's dad was a sniper in vietnam and he is a heck of a good shot.....For long ranges (im taking over 400) he always likes to use his 7mm reg mag or my 300 RUM (but he doens't really like the recoil that much). Just while ago(about a month ago), i got my 300 RUM in rem700 police, I thought I would be a better shot with this rifle than my 30-06..... boy Was i wrong, at 300 yards I CAN SHOOT BETTER with my 30-06 than my 300 RUM (why ?one, I think its because I know my 30-06 better, two I am less "shaky" about the recoil from a 30-06, three I practise alot more with my 30-06)
Bottom line its not your caliber, its not your gun, its not your scope (even tho some might say it is).... its REALLY all you, its all about how well you know ur gun and how it behaves and that my friend, takes alot of practise