For a hundred years the standard twist in 30-06 rifles has been one turn in ten inches.
They shoot every weight bullet from 110 grains to 220, or maybe even 240.
I think if someone wanted to shoot 150 grain bullets with extreme accuracy, they would order their 30-06 with a 1:12 barrel.
Standard 30-06 rifles seem to shoot heavier bullets more accurately than lighter bullets. Try some 220 grain in your rifle, for accuracy.
You can easily test your rifle, to see what the twist is. With a cleaning rod that does not swivel at the cleaning patch end, put on a patch that turns the rod as it is pushed in. When the patch is engaged to turn the rod, mark where it enters the rifle, and further back on the rod, mark which way is up.
Slowly push the rod in until it has made one complete revelution, then mark it again where it enters the rifle.
Push the rod through and the distance between your two marks will be its twist.
If you don't have a suitable cleaning rod, Canadian Tire sell wooden dowels of various sizes. Get one that just fits in yur barrel. A cloth patch over the end will make the dowel turn as it is pushed through the barrel, and the wood makes an excellent medium for the marks.