I have a 350 that was rebuilt about 15 years ago (bored .060 over) don't smoke or anything but seems like it just ain't got the compression it should have ,so in saying that ,when I check the compression and it ain't up to snuff should I just go ahead with new rings or trash it and start ove?  

new rings.  

go to a jukyard and get a new block.. they are cheap and sense you are .060 over the block is pretty much done...  

what is the block cast #'s?if you have one of the 010 blocks or even if you don't i'd take the block apart and check everything.one thing i've learned is don't trust the work done on your engine if someone else did it.everyone makes mistakes and if my motor screws up it will be me thats resonsible for it.the engine you have or any that you might get could have mistakes on it or something wrong with it so start with the one you have.after sitting that long the cylinder walls might be rough and need honing and new rings.i would treat it just like any rebuild.get some plastiguage and maybe go to sears and get a cheap caliper,feeler guage aand take a week or so and do it right.if you don't have the knowledge i recomend getting a book for about $20.00 by S-A DESIGN called,how to build the small block chevrolet,by larry atherton & larry schreib.it will take you start to finish the right way.  

What exactually do you mean buy low compression? What cam it has will effect cranking compression.You could have a brand new block,pistons,and total seal rings but with a cam of duration 280 deg. adv duration you may only see 145 psi . A vacuume gauge can be a good diagnostic tool if you have the litertature to diagnose it. Or a leakdown tester. I hate the idea of a complete rebuild if not necessary.I dont like the idea of .060 or more withoud sonic testing the block so if it is ring seal issues and there is a lot of taper in the bore you may as well scrap the block.  

Ball hone it and stroke to a 383. Keep the compression low and radiator full. Enjoy.