For a sweet, fast-in-the-hand deer rifle among the surplus ranks, the very best would probably be a Swedish Mauser or a Greek Mannlicher-Schoenauer. Both are light in the hand, extremely accurate and have sweet 6.5mm bullets that do a number on deer in SP configuration. The down side is theyr'e rather $$ and the Greeks are very hard to find these days.
Mosins, Mausers and Enfields are much easier to get but are going to be on the heavy side. I'd go with a 91/30 Mosin if you want an inexpensive one, and load with 180 grain Winchester/S&B SP's. That should work fine.
.45 Guy's suggestion of a Carcano is also interesting. Those are quite cheap, but better than their reputation would suggest. Norma makes quality SP ammo for them.