Hilton Head Island would be a great location to hunt. There are a lot of tourists who vacation there, but not so many detectors constantly scanning the beach like in Myrtle Beach. I have never been there either, but should start making my way down the coast checking out the beaches.
As far as Civil War relics, they are definately all over SC, especially in the Charleston area. As in all detecting, you can luck into a campsite or battlefield, but with a little research, you can place your detector in a field you know to be a good hunting spot. Research!
West coast or East coast? My opinion only: West coast is rich in buried and hidden treasures. Much more than that of the East coast. East coast is rich in Relics, with a handfull of buried and hidden treasures. Don't fall into that lost or hidden civil war payroll stuff. There are other treasures that are more likely to exist if you take time to do the research around the east coast.
Good luck, and if you pass through Myrtle Beach on your way through SC give me a shout.
xXx