I hunted deer a few years ago with a 308 110 gr Vmax loaded down a bit for neck shots. Think it was pushing about 3000 FPS.
Also the 110 gr TTSX barnes bullet is a good one and they also make a black tipped TTSX bullet for a 300 blackout that should be great if you want to down load to say 2500 or so.

Probably the hardest hitting low recoil round I have used is the 6.8 SPC. We have hunted hogs and deer with it and its a good round.
That is what my 92YO dad now uses and he loves it.

I load 3 bullets. Either the Raptor cutting edge brass bullet (expensive) and that has shrapnel that shears off the bullet and drops animals almost in their tracks, or a 110 gr pro hunter which is good for deer and small hogs, or the 95 gr TTSX barnes bullet which is a great hog bullet for more penetration.

I also have loaded the 95 gr TTSX bullet in 270 win to 3700 FPS and it makes a real mess.

My daughter for some reason decided to buy a 270 win after I told her to buy a 260, but any way she was complaining about the recoil.
I loaded her some 95 gr TTSX downloaded to about 2800-2900 FPS and thats a pretty mild round with a lot of kill power. The 82 gr ER Raptor bullet drops them faster but is more expensive.