The National Park Service lists a John R. McNeese/McNease enlisting as a private and attaining the rank of sergeant in Company E of the 38th Mississippi Cavalry. This almost certainly pertains to the 38th Mississippi Infantry (Mounted). Company E of this regiment was known as the White Rebels, raised in Lawrence County. Does this match your family? There are no other McNeese men listed. Here is a history of the 38th: