All the cartridges suggested above will work well. There are also a TON of other options that haven't been mentioned as they aren't as popular, tougher to find (the right) factory ammo for, require fire-forming of brass or aren't thought of as 200 yard cartridges. Hand loading opens up lots of doors. Do you like to stick out and be unusual or do you want a 308 like a couple other hunters at deer camp? Common cartridges will also let you bum some ammo when yours gets lost. Do you like to use historic cartridges, maybe give them a modern twist with non-standard loads? What kind of terrain do you usually hunt in? is it bushy or forested? What do you think about recoil and muzzle breaks? lots of questions to consider if you really want to dive in deep.

30-30 will easily reach 200 yards with the right hand load and bullet as will all the cartridges based on 30-30 brass that I have researched. The encore is capable of much more then 30-30 brass pressures so you are leaving a lot of performance on the table with those choices, but recoil will be minimal even without a break or porting. I am currently playing with a 7-30 waters load that will reach 200 yards on paper, but based on minimum velocity for the bullet I'm using, that looks to be about as far as it will go.

Many straight walled cartridges will reach 200 yards in a encore setup with the right bullet and load. 444, 45-70, 460 come to mind. (I would think a 357 Max would too. I know there are folks here who have shot 357 mags at much longer ranges.) The larger bore cartridges will have a good bit of kick without a muzzle break if you want to reach that far.

In the end, figure out what turns your crank, use it for a bit and then sell it to try something different if it doesn't keep your interest. Or keep it and buy another encore barrel, it's a well known fact that encore and contender barrels like to travel in herds and are lonely if there are less then 6-12 in a gun safe.