In my experience the 30-06 works best with 168 an up for bullet weight, I have loaded for many of them and I generally get the best groups out of 180gr Nosler BT's or more recently the Accubonds, I also find that the Hornady bullets give acceptable groups. I have one that will cut groups down to one ragged hole with 220gr slugs and they are deer takers for sure. I gave up on Barnes bullets as they are vary particular when it comes to how far you set them off the rifling and a very tiny adjustment can go from 1/2" to 4" in my testing. I have not found a deer or bear yet that would not fall to the BT even though they are fast expanding they hold together enough for black bear and deer and dump a lot of energy. Every shot has been a pass thru out to 300 yds.
If your only using factory loads I would try a box of Winchester 180gr silvertips or Federal Premium BT's. I'm having some great results with the new Federal Fusion line taking deer with my 223 AR's with sub MOA accuracy so that might be another round to try.
You will more than likely find your barrel is 1 in 10 twist in the 30-06, generally 308's are 1 in 12 and shoot better with bullets in the 150 range but there are always exceptions.
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