Ms. Brown-
Co. "D", 8th Tennessee Mounted contained a number of men named White. Since most units were locally recruited some of them were probably relitves of David Frank. here is a list:
Co. "D"
Herchel A. White
James C. White
John W. White
Stephen S. White

Co."C" contained the following
Clairbourn White
Samuel G. White

Co. "B" contained
John H. White

Perhaps you can connect some of these to your family, Good Luck