Seth J. Green is on the list of soldiers aboard the Sultana in the book "The Sultana Tragedy" by Jerry O. Potter. He is listed as a Pvt. in Company K of the 9th Indiana Cav. Also on Page 124 on this book there is a list of individuals who were selected to recieve money from a benefit staged by the Chicago Opera Troup and Memphis New Theater. Among those named: Seth J. Green 9th In Cavalry, $10.
You can request information from The National Archives in Washington D.C. that might give you the info you request.