Well aside from a .308 battle rifle like an FAL or AR-10, the selection is kinda limited.
still makes the 7400 in various trims and calibers. The one I have in .243 is a horrible shooter, but if it's an upgrade from a Mini-14, probably never notice...
The only other big bore self loader I can think of is the Browning BAR.
I've never even held one, but I've heard they are better shooters than the Remington. They are also pretty heavy.
I guess if you're hunting close, a Ruger Deerfield Carbine
counts as a hunting rifle. The original version was discontinued because of durability issues, and I have't heard about the new one yet. It's also going to be limited to about 150yds. tops. What and where is he looking to hunt?