I have located a letter sent to my father by his aunt Amelia Pulliam (who was my great aunt) in September 1948. The letter includes some genealogical information. I am posting that information here to help anyone else trying to trace that line. My only documentation is this old letter, but perhaps some of Amelia's details will offer enough guidance to help another person make a Pulliam connection they have been searching for.
Amelia writes the following to her nephew (my dad) -- all the relational terms she uses are in relationship to him:
Great-great grandfather John Pulliam lived in Oxford N.C. buried in New Port Tenn. burial ground in the plot belonging to the Roadmans, whose daughter was wife to R. W. Pulliam who is buried in Riverside here (she was writing from Asheville NC) in the same plot where Amelia's parents rest
Great grandfather R. W. Pulliam 1st (Robert William(s) Pulliam, who must have married a Roadman)
Grandfather Lawrence Pulliam (Amelia's father) who Amelia says had sisters Mrs. McLoud "aunt Ella" and Miss Mary Pulliam "aunt Edie"
Father Lonnie Rickey Pulliam (Amelia's brother)
Robert Williams Pulliam, Amelia's nephew and my father.