I'm a descendant of James & Ellen Hunnicutt by their son, William John Hunnicutt.
This is the first I've heard of the Hunnicutt Burying Ground "going" private. Please contact me with the details. I still visit the "Old Country" to visit relatives in Oconee and Pickens Counties. And I understand that James's parents, William and Elizabeth Hunnicutt are buried in that cemetery.
I have a copy of the Hunnicutt Family History by Jessie Hunnicutt Burgess. In it, I find "Rudie Caroline Hunnicutt married to E. A. Mulkey. They are buried at Hunnicutt burying ground, Salem, South Carolina.)"
Rudie is listed as the daughter of James Madison Hunnicutt and Ellen Swafford Hunnicutt. I have a bit more about her, all from the Hunnicutt Family History Book. I have verified very little from that book, and most of the information in it has no documentation that is listed.