Beside the beach and the bike paths that are all over Hilton Head, there is an off-leash Best Friends Dog Park at Chaplin Park that is open sunrise to sunset. It is fenced in, tree lined (mulch and leaves on the ground) and includes benches for the humans, water fountains for the dogs and a fenced area for smaller dogs. We do recommend you bring bug spray and water for humans but there are restrooms close by and free parking.
There are restaurants that are also pet friendly (mostly at the outdoor seating) and there are new ones that allow them every day, so to be sure call the specific restaurant and ask. Here are ones we know of:
There are also some attractions that allow well behaved dogs on a leash to join in the fun.