It appears that you are asking about three different men named Oliver and one named Harrison who belonged to the 6th Kentucky (CSA) Regiment. You can order copies of their military service files here, as you will note at the top of the page. History of the famous Orphan Brigade includes the Battles of Shiloh and Baton Rouge, which took place months before Murfreesboro.

http://www.rootsweb.ancestry.com/~orphanhm/history.htm

The term 'mounted' does not applied to these regiments before the summer of 1864.