Most 30-06s will shoot 165 gr bullets well compared to equal quality loads of different bullet weights.
What exactly are you not impressed with in the Remingtons? Accuracy? Terminal performance?
What specific Remington bullet is it?
If you don't like the accuracy then try a higher quality line of ammo in 165 gr.
Some things that come to mind are Federal Premium, Winchester Supreme, Hornady, Black Hills Gold, etc.
Those lines of ammo are loaded with bullets that are typically more consistent (read accurate) than regular Rem or Win soft points. These are bullets like the Nosler ballistic tips, partitions, accubond, interbond, fusion, etc.
The Ballistic Tip and similar bonded bullets with polymer tips tend to top the accuracy list.
As far as terminal performance goes it doesn't really matter what bullet you pick. Deer are not hard to kill. You're going to kill a deer with any bullet out of a 30-06 whether it's a $14 box of cheap Winchester soft points or a $50 box of Barnes TSX. If the bullet doesn't kill the deer then it's your fault.
Having said all that, I used to have a Win Model 70 in 30-06 and used 165 gr Federal Premium Ballistic Tips. The Ballistic Tips killed deer just the same as Winchester Super-X soft points or Barnes TSX. The Ballistic Tips just happen to shoot better so thats what I used.
You simply have to try ammo until you find one that works for you.
Higher quality ammo almost always shoots better and 165 gr bullets typically shoot better than 150 or 180. You should still try some 150 or 180 gr loads if you don't mind plopping down the cash for some expensive experimentation.