I don't know if these railways were Confederate or Yankee controlled at the time frame you are looking, but according to the book The Railroads of the Confederacy by Robt C Black III this was the direct path from Chattanooga to Gordonsville VA.

E TN & GA (from Chattanooga) to the E TN & VA to VA & TN to Orange & Alexandria to Virginia Central (Gordonsville VA) (5 different railroads)