I carry a savage bolt rifle that holds 4 rounds in the magazine and I Carry 1 spare magazine of 4 rounds in a pouch on my belt. Again, not so much to shoot it out with the elk, but on the off chance you need to shoot more than twice or have an ammo failure/etc.
Before I had a clip rifle I used an uncle mike's slip-over bandolier on the stock and would carry 4 in the rifle and 8 on the stock (because the extra wieght made the gun balance better) but with heavy handling your bullets CAN fall out. Don't store the rifle with the bandolier on the stock, remove it so that it stays "stretchy" over time.
My dad hunts with a 760 Pump gun and carries two spare clips, but i think this is a habit he picked up antelope hunting.. where we shoot at much longer ranges (and have been known to run back to the truck for more ammo)
For heavy cover hunting/signaling/finishing off/etc etc etc. I carry a 44 magnum and with and 18 rounds on my gunbelt. Its a lot of extra wieght but I'd rather have it than not. (before i carried an elk-legal handgun I carried a .45 and 3 clips.. about the same amount of ammo).
When I carry my pump shotgun with slugs I'll put 4 in the gun and put an unopened 5 pack of shell in my pocket (they don't rattle).
Better to have a hard case than a soft one. 1. It will protect your optics better.
2. It will protect your finish better
3. You are less likely to leave it laying around by accident and lose it.
The exceptions to this are: a "parachutsists weapons case" Designed for holding your long gun while you jump out of airplanes, you can also unfold it and use it as a shooting mat. Or if you have iron sights, a heavily padded BOYT case is nice. I like the ones with an accesory pouch for ammo/spare clip etc.
Hope the info helps,