Well I guess I won't have to tell you to get a 243 (LOL).
The progression goes as this, the 308 is a modified 30-06 case, the 243 is a modified 308 case. The 243 by the way is a 1,000 yard round if you want to load them that way.
While you will get accuracy from a 308 for about 5,000 rounds, I should warn you that the 243 will only get accuracy (sub-MOA) for about 800 to 1,500 rounds if you use hot loads (3,600 fps to 4,000 fps).
Bullets for the 243 come from 55 grain to 107 grain weights. What you want for deer is anywhere from 85 grain to 107 grain bullets (there are a ton available).
PS: your shoulder will thank you since the 243 is a mild recoiling rifle (you can shoot it all day long).
With proper shot placement the 243 will drop a deer in it's tracks. The shinning thing about it is the Hydrostatic Shock it produces because of the speed at which it travels. It is flat shooting, soft recoil, easy to reload, super fast (just don't use slow commercial ammo) and has a ton of options (varmint as well as big game).
Congrats on your choice, you will be very happy. (or so I think)
Jim