480: Go with an Encore in the 308 caliber. I am not a wild fan of the 308 but it will do what you are looking for your hunting handgun to do, and not break your wrist in the process. The 308 cartridge will reach that distance on the game you identified.
Since you already reload for the 308 you are one step ahead already. Also, in the Contender/encore in 308, at those ranges you would not need expensive bullets to do the job, your handloads will do it just fine. My brother has one of the Encores in 308 and the loads I work up for him, right from the Lyman 45th edition, cluster one hole at 100 yds, whether 150 or 180 grain. In addition, at the distances you mentioned, round nose 180s might even be the most accurate bullet for you.
Further, using barrel inserts for lighter caliber cartridges with a 30 caliber bore diameter (such as the 7.62x25, 30 Luger, 32 auto) , or light handloads using cast bullets, you can shoot or practice all day enjoyably. Hope this helps. Buenos Luck. Mikey.