muddy
If you are going to use it to varmint hunt a little too, then 243, if deer only then 7-08 or 308. Recoil gets sharper as you go up, if that is a concern.
Not sure how hard ammo is to find on the 7-08, but around here it's kinda scarce. 243 and 308 are common.
The 243 Handi's seem to like the lighter bullets, like 85gr or so. Some shoot the 100's but most don't.
If you don't reload, and ammo is hard to get, buy the 308 and don't look back. Ammo everywhere in lots of weights. It will even serve for heavier game than deer (can you say elk?) with a good bullet.
A 165gr Nosler partition at about 2700 is good on elk to 300yds, so I know it will do deer nicely. Less recoil than an '06 too.