I've seen that Pettypool name among other writings of Carolina Pools. The name was shortened sometimes on ship manifest like "P'Pool" and may have evolved to "Pool". I know about the Jamestown Pools because my family elders handed down thru the years said Robert Pool was our first Pool, and the name Pool came from Engalnd then evolved to Poole just in the last hundred years. The Jamestown Historical Society website may give you info on Pettypool, I don't recall seeing that in there. I just suggested your William, born in Elizabeth City Va may have descended from Robert because he had a lot of "unknown name" children and owned propert there.