It's quite possible his father was conscripted in 1863 into the 4th Texas Cavalry

James McCune, Private, Company B, 4th Regiment Texas Cavalry, absent sick October 24, 1863 remaing so through the end of year, recorded absent without leave on the February, 1864 Regimental Return, no further records

The 4th Regiment Texas Cavalry was organized about October, 1861. Also known as the 1st Regiment Sibley’s Brigade Texas Mounted Volunteers, and as the 4th Regiment Texas Mounted Volunteers

Note, The last Company B muster roll was dated January 31, 1863 which would indicate James enlisted sometime later. Most likely Conscripted. With no further Company muster rolls in the records, there is way to tell if he later returned to the regiment

M323: Compiled Service Records of Confederate Soldiers Who Served in Organizations from the State of Texas

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FOURTH TEXAS CAVALRY. The Fourth Texas Cavalry, also known as Fourth Regiment, Texas Mounted Volunteers, was organized in September 1861 and was assigned to Henry H. Sibley's Army of New Mexico. The men were primarily recruited from the counties of Caldwell, Cherokee, Dewitt, Guadalupe, Houston, Milam, Nacogdoches, and Victoria.
http://www.tshaonline.org/handbook/online/articles/qkf18