hmmm... this will be actually tougher than usual; the handloading thing throwing everything about here. I would have said 22 hornet; which is proly the handloaders dream cartridge, but you said to hunt possibly as well.
now then, if you paper punch, you certainly can do the 243 or 25, and they do have a nice variety of bullets; especially the 243, you can get some very nice vld type bullets for it. As far as bbl burners go; I say mmeh! I suppose, if you over shoot them, don't let them cool down at least 1 minute between shots, shoot hot loads, etc.
I have an old, OLD OLD; remmy 788 in 243. Very well worn, you can tell by looking at the bolt. Yet it still shoots absolutely lights out with factory feddy ammo blue box, the cheapest stuff you can buy.
Will it win a benchrest competition? proly not. Could it? maybe on a good day, with a great shooter. I am pretty good I think, and I can get it sub moa at 100 yds, every time I shoot it. Will it kill a deer or pig? Absolutely. I have put many a round through/behind the shoulder of a piggie, and it shears off the tops of the lungs, and heart usually, and they end up, seriously doing the kickin chicken. I am trying to think more of a practical standpoint for you here , though. As in cheap- just my thinking not trying to imply yours.
You could do better with a 260 remington, like a better 243, and components will cost the same. Also a very nice round would be a 6.5 swiss; very nice and mild. Also what comes to mind is a great round in the 6mm area; the 6mm remington. It is also a faster 243, but with the same bullet. Now then, people will tell you it is a barn burner, but I say no more than a 243, because of it's long , elegant neck/case length, the brass helps a lot to cool down that burning gas plug, before it gets into your bbl throat.
Let us not forget some other great 6 class rounds, the 6 ppc, the 6 xc, and I would also not overlook a 6.5 lapua or 6.5 x 47, these last ones are very mild on recoil.
Of a matter of course, i like very small stuff that does a big job, or carts that really max out there potential, without a lot of firepower, so I will throw in a couple of others to think about here. a 6.5 grendel would be great- a small , fat case, which we know does a great job in cooking the ingredients, fireing out a greatly accurate round. and last but not least, a 7.62 x 39.
there is a dude here , who handloads a savage rifle he had built in this cal, and it wreaks of fantastic accuracy. plus the cartridge does a big job, with a little powder, and bullets and cases that don't break the bank.
CZ makes the little 527 carbine, which I have, which can turn even the crappiest of eurotrash 762 rounds, and make them pretty darn accurate. And it will have set trigger as well, yummy!
It is a nice looking rifle, and fairly expensive, expect to find them new for 600 or so, used you might find cheaper. Again, it is nice, with great fit and finish, and very nice wood, so if you want a beater, get a savage, get a bbl made for it, and make it into a long bble'd tactical, that you don't mind getting dirty.
I am going for a max here on recoil by the way; i am saying the above carts, will not eclipse the felt recoil of the 25. this is why i did not mention stuff like a 7.08 or such, though a 257 roberts could do the trick as well, but then again, that is very close to a 260 remmy, so why bother, unless the components are cheaper...
But hey, I have gone on way to long here, so lets hear some other dudes.
Again, my priorities were to keep the component cost down, and keep the recoil down to the 25 level,
or less.