Kind of similar shooting to you. My local range goes out to 200 yds. My next closest range 45 min. away goes out to 300 yds. I have a Rem. 700 in .223 that I regularly shoot out to 300 yds and can shoot 1/2 MOA 5 shot groups when doing my part. Around 2 hours away I've discovered a 600 yd range I'd like to try (haven't yet). As others have said, a .223 with heavier bullets (I load Hornady 75gr. BTHP's) certainly gets to 600 supersonically. The .223 I've put somewhere north of 700 rounds through and am totally happy with.
Like you, new toys do have a certain appeal. I recently came across a Rem. 700 ADL in 6.5CM that after sale pricing, Remington rebate and some Cabela's points, came in at under $200. Couldn't pass it up. So now I'm building a 6.5CM (for the reasons other have mentioned) similarly to my .223 for "mostly" shooting up to 300 yds. Do I need a 6.5CM...nope. Do I want one...yup!
Based on the road I've traveled, I would recommend starting out with a .223 with at least a 1:9 twist, if not a 1:8 twist so you can shoot heavier, higher B.C. bullets out to 600 yds. You can certainly trick it out to make it yours. It doesn't have to be run of the mill. On mine, I plopped it into a KRG Bravo stock to get proper fit, added a Timney 510 trigger and a Vortex PST II scope. Pretty cool rifle.
Once you dial in your technique (breathing, pulling trigger straight back, loading a bipod, follow-through, etc.), maybe consider a longer distance caliber. Some low recoil, high accuracy, not crazy expensive options (again, as others have mentioned) are .243, 6.5CM, 6mmCM, 6mmBR (if you reload), 260 Rem, .224 Valkyrie. I've shot a 7mm Rem Mag. (last week actually) and a 300WM, and for punching paper or steel out to 1,000 yds, I don't know why I would get one. A 300WM for instance was kind of an interesting idea until I shot one - yup, no need for one of those! Heavy recoil, expensive rounds, go through barrels quicker (although the diminuative .243 and 6mm CM can also be barrel burners) don't help you land more hit's in the x-ring out to 1,000 yds. Just one man's thoughts. Let us know what you decide!