I would like to know what your load is for doing that. You've got 125 fps on a Nosler factory load and 100 fps over the A-square load.
The extra 30 grains and 100 fps isn't enough for me to bother with the 338-06 again. I'll go straight to 9.3.
I've found at least 10 250gr loads that run from 2569 to 2645 so it is doable.
Rifle #105 has 6 loads in that range.
Hornady 5th has 2.
Reloadersnest has 2.
Handloader #184 has 1.
I don't get 2600 with my current 19" 338-06 but 59gr of IMR4350 under a 250SPBT gets me over 2500 with no pressure signs. 60gr H4350, same bullet, gets me just under 2500.
The highlighted portion of your post is where we differ. I don't use, nor like, any .30 caliber so the jump from my .284 to my .338 is .054" which is similar to the gap from .308 to .366.
