I have several 06's, more than I care to admit. they are 2, 4, and 6 groove. All are 1-10 twist. Barrel length is 24-26", one is a bull barrel, the others military or newer off the shelf Remington 700.
Here's a couple to try;
180 Gr Sierra Game King
43.5 gr. Varget
Oal 3.30 if your have the room
Should get you around 2500 fps.
Spendy but I get excellent results from two 06's and these are strictly for hunting so I don't need a lot
.
Swift 180 gr. Scirocco ll
55 gr. H 4831sc
.02 off the lands in my guns.
Both of these produce less than an inch groups at 100-115 and do so consistently in my 1-10 twist 06's.
I have been told and read more than once that 4831 wasn't an ideal 06 powder. I have used it for years with nothing but success and it's my go to powder from 150 gr up to the 200.
A fun load that seems to shoot amazingly well, some really tiny groups is;
125 gr. Hornady sst
48 gr. IMR 4064
2800+ fps.
I seat these just short of the crimp groove.
It has very little kick so it's great to just have fun with.
All loads are in the books and below max.
Good luck