Hello Huntin Junkie and welcome to LRH,
The question of range is dependant on a lot of variables. The short answer to your question is that 880 yds (half a mile) is a bit far for a standard 308 shooting typical bullet/load combinations.
First thing to consider is bullet performance. Most "hunting" bullets expand down to a velocity of 1800 fps. Some a little higher and some a little lower. I dont tell anyone else how to hunt but I use the performance limitations of the bullet as my hunting range limitation in most cases, especially on larger game.
From there it is only a matter of running the numbers on a ballitics program. The main factors that will affect the range of the bullet are BC, velocity, elevation and temp. Temp can also affect velocity by affecting the burn rate fo the powder. Barrel length will also affect velocity and every rifle will shoot different than another.
So if we look at a typical 308 shooting a 165 Nosler AB with a BC of .475 and a generous MV of 2900 out of a 24" factory barrel @ 1000' elevation, you can get as far as about 630 yds with 1800 fps of velocity. If you use a 168 AMax, you can push that out a little becuase it opens at lower velocities, but with those weight bullets you are also running low on energy and momentum. If you jump up to 5000' you can stretch it out another 90 yds.
Some powders might get you a little more velocity, one of which is RL17 which works well in 308's and a lot of other cartridges. I have seen boosts of 200 fps using RL17 in a 300 WSM and a 25-06. So with the right bullet, in the right rifle (26" barrel or longer) and the right powder, you just might be albe to stretch to a half mile, but it all has to come together.
Being interested in LR, my choice in the 30 cal would be the 300 WSM. The rifle will cost the same for the same model. The differnce in reloading costs will be very minimal and you can shoot higher BC bullets faster. The difference in range will be about 200-300 yds. And the 300 WSM will last a long time.
The draw back is that it will have more recoil, but there are trade offs and costs to pay for performance.
I personally dont like brakes, but that's a personal choice. If you're looking for a carry rifle, you'll need to get your ear plugs in before shooting, so quick offhand shots will be very difficult.
Hope that helps and good shooting,