22-250
Consider reloading the .308 and with that consider reloading with some of the newer sabots and .22 caliber target bullets.
Although I used to shoot the 22-250 rather well on ground hogs out to a little over 1,000 yards with a bipod and scope, I found that I could shoot the .308 sabot even more accurately and that I then had a great rifle with a huge lattitude of loads from ultra long distance varmit to big game. You could do the same in 30-06 as well.
Just a thought. I don't reload any more but I imagine that the sabots are still available and probably even better ones and probably even better bullets than I had.
According to your criteria and if I were you... I'd love to have a .308 for the cheap surplus ammo, the wide selection of factory made ammo and reloading supplies AND it would be great if they made a .22 or smaller bullet and sabot for it with about a 70 grain bullet -- that is long and skinny...
I don't shoot such a gun any longer due to spinal problems and shakes but if I were younger that's what I'd be lookin for.
Since you may well consider reloading anyway because of the distances you plan; you might want to consider a single shot such as the Thompson and some unique round. As a rule, according to my limited experience, a long skinny bullet in a sabot with a large cartridge is FUN for what you are doing. One thing that is available now that was not when I shot such things, is the wonderfully accurate range finders. With these range finders now, the correct distance and therefore the absolute correct elevation corrections allow for much better judgement in ultra long distance shooting. Last year we checked one of the new and expensive "laser" range finders on a measured mile and it was only 12 feet off!!!!!!!!!!!!!!!!!!!!!!!!!! That's good enough!
But there are many on TFL with current experience that can advise you better.