The .30-06 will handle a wide range of bullets.
When deer hunting, I like to carry a Hammond Game Getter -- this is a .30-06 case with a steel base (you can get them for other calibers, too.) Instead of a primer pocket, the head is chambered off-center for a .22 nail-setting blank. With a buckshot in the mouth, it makes a fine small game load. Using #2 blanks (brown) and a swaged 00 buckshot, I get around 700 fps. It's a real cat-sneeze load, but groups inside an inch at 25 yards, just at the top of the lower thick crosshair with my scope set at 3X.
I like cast bullets -- my preferance is for a 160-grain cast Lee lubed with liquid Alox and driven by 22 grains of H4198. I also like Ed Harris' "The Load" -- 13.0 grains of Red Dot behind any cast bullet in any .30 caliber case of .308 capacity or greater.
If you're into "store-boughten" bullets, you can get 100 and 110 grain "plinkers" and load them from 1,000 to around 2600 FPS with fine accuracy (some people go even higher.)
There are now 125 grain bullets designed for "deer loads" (as opposed to bullets designed for varmits) which allow you to assemble reduced-recoil hunting loads (of course, you can buy "managed recoil" loads across the counter with the same bullet.)
The saboted .22 caliber, 50 and 55 grain bullets look intregueing, but I never met anyone who could get the accuracy necessary to make them really good varmit loads. I'd go with a 125-grain varmit bullet if I wanted to use my .30-06 for that purpose.