6.5-06 sounds like an easy one to do on your rifle too --same bolt, same action, same mag, etc. Otherwise, for long action, the .300WM isn't a bad one to look at. I used to have a 700 PSS in .300WM and it sure had some reach. But regarding .308, anything 6.5 will outperform that ballistic-wise, particularly when given even the same or better, a larger size case! .300WM can take advantage of heavier bullets with much better BC than the .308 can, so if you want .30 cal. bullets with long range, this is the ticket. And yeah, it'll last a lot longer than .338 or any of those super long range calibers will. It'll cost a lot less too. I got into fifty, and let me tell you, that's an expensive one to get started in. But after initial cost, the components aren't THAT bad, considering. Components for other long range rifles cost MUCH more.
You also have the 6.5 Creedmore, real popular, 6.5x47 Lapua, .260Rem., etc. And 6.5-06. As for modifying brass, modifying .30-06 (and maybe even what you have) into 6.5-06 will be easier than you think. You can go on the cheap on components this way, as it's that rare brass that kills you on cost. You'd be able to use all the high BC and heavy 6.5 pills, and with that case capacity, be able to way outperform, I'd imagine, any of the smaller 6.5's out there.
And all you need is a good set of dies, a good trimmer and caliper (which you likely have already) and a nice, new 6.5-06 bbl. of your choice. If you were gonna handload anyway, this would be my pick.