One of the best double duty calibers out there for light skinned, deer sized game and varmints is the 6MM/.243. There are many ways you can go with it also from strictly commercial loadings such as .243Win, 6MMRem, and .243WSSM to a large number of full custom, hand loading only, 6MM rounds. BullBerry alone lists 14 6MM/.243 chamberings and I can think of another 5 smiths without much trouble. Since you are shooting an Encore you are free to choose just about anything you wish from anyone including the JDJ customs.
Things you need to consider are; cost of shooting. A long session over a PD city can really burn up powder and at $10 per 20 for plain Jane .243Win that could get expensive real quick. I use the .243Win because that is about the least expensive commercial ammunition out there. There is also the cost to you physically. Most of us hairy chest, shooters like to think that many hundreds of rounds of .25, .30, or even larger caliber rounds at a session wont effect our shooting, well it will, and it does. The .6MM/.243 is about as small/big as you can get to cleanly kill deer sized game, and shoot long varmint sessions and still be able to use your arm to drive yourself home. There are things you can do to dampen out that recoil from the bigger calibers but it seems to me the whole idea behind the TC/Encore is handiness.
As for a pistol barrel you cant go wrong with the .308Win or the 3006 for strictly commercial loadings but Id opt for a .257Roberts or .25WSSM some commercial loaded stuff but mostly a hand loaders game.