I need some recommendations for a good off the shelf hunting rounds for a FN Patrol 308.

Define good? What are you hunting? The best 308 ammo will be hand loaded for consistency, higher velocity, and accuracy.

The following are simply my narrow conclusions from my narrow experience when I got my 308. It's a 26" shilen barrel on a savage.

Federal Gold Medal Match with the 168 or 175 Sierra matchkings shot very well. Its what I would probably buy if I was hell bent on new brass. just started to challenge myself to shoot at a higher proficiency level and I thought both these were excellent value to get the accuracy level I was looking for. But even buying in bulk meant over $1 per round. I planned on shooting thousands.

The ammo I'd probably start shooting the most is Chad's 168amax, the more economically priced one with the once fired brass. I'm totally not saying that just because he's here. I shot a few hundred and there was just small difference between it and the federal gm in my rifle. But for the price difference of sometime 30-40%, hard not to like it. It was easily 1moa out to 300 nearly everytime and I assume operator error usually when it wasn't.

Ultimately I started reloading but I lve shot a few hundred of both and had great luck with both.

ChadTRG42 ,

I would like a round that would hold an inch or less group and be able to take down any of the North American game animal with some of the common Texas Exotics.

Korean Redneck ,

What do you think of the 168 or 175 Sierra matchkings as hunting rounds?

Personally, I wouldn't shoot a thin jacketed bullet like the MK in most hunting situations. Try the GameKing if you like Sierra Bullets. Or get Chad to load you up some different test rounds with different bullets and see what your gun likes best. For off the shelf hunting bullets, my 26" rifles like Federal Fusions. But they're not going to give you sub 1/2 moa results.

Sorry, when I hear 308 I always assume punching paper and hitting steel. Never tried as my hunting experience is more limited. However, I don't think I'd try it with any bthp.
But then again I have a different approach to hunting ammo.
Since I'm usually fairly comfortable with my 308 and if i were to hunt with, then I'd actually start with winchester silver box and/or remington core-lot. You never know if one of the cheaper ones end up shooting well enough (let's say sub -moa at a range). It's rare but you might get really lucky and find one that your rifle likes.
If not, I'd look more at the bullet than the ammo manufacturer. In my experience, and I'll through Chad/Dallas Custom in the following mix, I wouldn't hesitate one bit to try a smaller manufacturer if they produce it with bullet you're looking for. I can be douche about hunting ethics, meaning I'd rather error on conservative side and give bullet selection some thought is important in doing so.
For example: if this were for a larger game animal (much bigger than white-tail), then I'd start looking for something like a Hornady 180 eld-x or 180gr Nosler partition or a barnes TSX bullet. Something on the heavier side and good, known bullet.
If it were more medium game, then I personally just like 150 or 165gr sierra gamekings.

Btw, my 308 is beast in terms of size and I have a 30-06 so I don't hunt with my 308, yet.

A couple of things... The Sierra Matchkings will kill just fine with good shot placement... BUT. These bullets will have inconsistent results on expansion and penetration. I've seen them ice pick through (no expansion) and I've seen them upset dramatically with rapid expansion and not much penetration.

For hunting bullets, any basic hunting bullet should do fine for Texas hunting (WT deer and pigs) along with a few exotics. You can get an expensive cup and core bullet, like a soft point, and they generally work well. You can get a Hornady ELD-X for controlled expansion with a decent BC. You can go premium bonded bullets, like a Nosler Accubond, that will give you 80%-90% weight retention and deep penetration. There are many options.

For me personally, it's hard to beat a 165 grain Sierra Game King (soft point, boat tail), 168 A-max, 178 ELD-X, 150 or 165 grain Nosler Accubond, Hornady 150-165 SST, and my favorite right now- 125 grain Ballistic Tip at 2900 fps in my 16" 308.I have used all of these with great results. The options are almost endless with a 308. If you want some match grade ammo loaded with the bullet of your choice, let me know. I shoot a TON of 308 and know it well.

My first choice for off the shelf all around hunting ammunition would be Federal Premium 165 grain Accubonds. I would also look at Chad loading whatever bullet you want but the 165 Accubond is pretty ideal everything bullet for a 308.

I roll my own now, but I've had GREAT accuracy with Hornady Custom 150 SSTs. Might go up a step heavier for the really big stuff, but 150s in a .30 cal is all I've ever needed.

Just buy Chad's bulk 308 ammo, it's consistent and has a good bullet in it(and it's cheap)

Hornady American Whitetail in 150 or 165s