I think the twist in most '06 barrels is not conducive to anything less than 125 gr. IIRC. I have shot 110 grainers at coyotes with success, but it took a lot of effort to zero it in and I had perfect weather - no wind, stable rest. The head shot was magnifique! Until I really have to time to work up a handload I'll stick to the 150 ballistic tips because I can get sub-MOA out to 200 yrds and head shots are the best way not to ruin to much fur.
I really would rather have a .22-250 for varmints than my beloved '06. At least I'd have longer, flatter range for some of those pesky critters.