You are correct, William H. Champion did reenlist. He worked at the Newson Hospital in Jan 1864 and was killed at the Battle of Resaca on 15 May 1864. According to what was repeated by his grandson (my great-grandfather), he was shot by a sniper at Resaca.
And he was the half-brother of Teresa Jane, his mother was Elinor. His son, Tom Champion, married Delila "Dealie" Hudson