recomended whitetail and muley cartridge.

[PJgunner]

I have never played with an Encore, so this may be more conjecture on my part than anything else. How heavy is the Encore? I have the feeling is is a fairly light weight rifle. Therefore, I would rule out the .300 Win. Mag. even though it is one of my favorite cartridges.
This leaves the 30-06 and .308 Win. which are, at least in factory ammo, just horses of a different color. I recently did a short test comparing 180 gr. Silvertip ammo in both the .308 and 30-06. In a 22" barrel, the .308 was actually faster. The 06 was a bit closer in velocity in a 24" barrel compared to the 22" .308, and finally, in a 26" barreled Ruger #1 the 06 came within 50 FPS of advertised velocity. I haven't had a chance to compare other brands with 180 gr. bullets yet, but will sometime down the road. So, the choice of either the .308 or 06 is push compared to shove.
However, if you handload your ammo, the 30-06 can be loaded up to be close to or equal factory .300 H&H ammo. You have to be quite careful though as loading for a single shot is somewhat different than for a bolt action. Most obvious pressure signs in abolt action don't even show up when loading for the one-shooters. I base this on handloading for 17 Ruger #1 rifles in various calibers. This is where careful case head measurements and a close eye on chronograph speeds can make or break your day.
Regarding the .308. It is more than capable of reaching out an touching something. Not quite 500 yards, but the two longest shots I ever made on Mule Deer were 250 yards here in Arizona and 427 Paces on a 195 pound buck in Nevada. I don't remember the load for the nevada deer other than the bullet was the 150 gr. Sierra Pro-hunter, although they didn't call them that way back then, and the powder was H-335. (It was cheap and it worked. ) The Arizona deer was shot with a 165 gr. Speer Hot-core over 49.0 gr. of W-760 for a velocity of 2550 FPS from the 18.5" barrel of a Ruger RSI. The same load delivers 2610 FPS from a 22" barrel, FWIW. There are betetr powders for the .308 than W-760, but that's just about the only load that very fussy rifle will shoot well.
Just one addition here. That long shot was at a 195 pound (dressed) deer than one of my hunting partners had wounded and was getting away. I will be the first to admit that it was strictly a "Hail Mary" shot. Lucky for me and unlucky for the deer, however the point is the .308 can do the job way out yonder.
I've hunted several of the western states most of which are quite open, and other than the two deer mentioned, most of the deer I've killed in California, Nevada, Idaho, Utah and Arizona could just as easily been taken with an open sighted Winchester 94 in 30-30. In fact the last two deer I've taken here in AZ were at 35 and 50 yards in fairly open areas. I used that .308 Ruger on those two.
Paul B.