Here are two possibilities:

William C. Womack
Enlisted on 6/1/1861 as a Private.
On 6/1/1861 he mustered into "E" Co. TN 16th Infantry

…..

William C. Womack
Enlisted on 11/29/1862 as a Private.
On 11/29/1862 he mustered into "E" Co. TN 23rd Battn Infantry

Sources used by Historical Data Systems, Inc.:
- Index to Compiled Confederate Military Service Records

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William Womac, Private, Company A, 16th Tennessee Infantry, enlisted May 18, 1861 at Camp Harris, Tenn. for 12 months, died October 20, 1861

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William C. Womack, 22, Private, Captain J. L. Moore’s Company, Newman’s Battalion Tennessee Infantry,* enlisted November 8, 1862 at Lynchburg for 3 years or the war, subscribed to the Oath of Allegiance at Tullahoma, Tenn. February 3, 1865, resident of Lincoln County, Tenn., dark complexion, dark hair, dark eyes, 6’ 0”, deserted December 10, 1864

* Also known as the 23rd Battalion Tennessee Infantry

Compiled Service Records of Confederate Soldiers Who Served in Organizations from the State of Tennessee