All Confederate military property reverted to the U.S. government at the end of the war. The C.S. Army formally seized Hunley soon after its arrival in Charleston in the summer of 1863, but (IIRC) never got around to paying Horace Hunley and his partners for it. Up to that point it had been a private venture.
I don't know specifically how South Carolina defines its "Waters of the State," but the U.S. Abandoned Shipwrecks Act of 1988 transfers all abandoned wrecks lying in waters claimed by the states to those states' jurisdictions, to be dealt with according to relevant state laws. Individual states' laws regarding historic shipwrecks and archaeological sites vary tremendously. In Texas, the Hunley and Housatonic sites would be clearly within the boundaries of state lands, and so protected under the Texas Antiquities Code as well as being claimed by the present-day U.S. Navy, but I don't know about South Carolina.