Personally, I would stick with .25-06 Rem. That's what I have, so I am partial to it, I guess. For moose in Ontario, I go w/ my .308; 180 gr. But I think .25-06 120 gr would take a moose without much problem (again, shot placement). My Tikka T3 Lite SS in .25-06 loves 115 gr Winchester BALLISTIC SILVERTIP rounds.
.25-06 will do for you what a .22-250 + a .243 + a .270 (up to 130 gr) would do. It is fast, flat shooting, and "kicks very little" (far less than .270 or .308).
I have no experience with 6.5x55 but from what I see online, it can't do for you what you do with .22-250 (varmint). But it does handle more weight, yet it has similar punch.
I compared .25-06 (117 gr) to 6.5x55 (140 gr) and the .25-06 beat it in both energy and trajectory (from muzzle up to 500 yards).
When I compare .25-06 (117 gr) to .30-06 (150 gr and 200 gr), the .30-06 has more energy at muzzle but it looses that energy a lot faster than .25-06. The difference is negligible. And I don't need to tell you about their trajectory.
So, I am sticking with my .25-06 ;-)