I have both 308 and 260 and have hunted for years with the 308... I just got the 260 and hunted with it last season. I use a 168 berger vld running around 2700fps in the 308 and a 140 a-max moving 2825 from the 260. Both are great calibers for deer. After using my 260 last season and spending about eight months with it developing a load for hunting and general target use at my 1000 yard range I see no reason not to use it. The 260 a much easier cartridge to make first round hits with at longer distances. It requires a wind hold about a third less than the 308 and uses about a twenty-five percent less elevation at 1000.
At 500, using my load data, it's also hitting with a little more than 100 foot pounds of energy compared to my 308 load. I have always used 1800 feet per second as my general rule of thumb for range limit as this tends to be the threshold for many bullets to expand. Using this rule in the past 500 yards was my cut off for the 308, the 260 can go another 200 yards (700y) before hitting this speed... Just something to consider.
Either one you choose will be good from my experience at 500 yards.