Yes, that appears to be correct, unusual that he would re-up!

Jesse A. Gladden age 26, Private, Capt. J. M. Moore’s Company (Pickens Guard), 6th Regiment South Carolina Vols.*, enlisted June 15, 1861 at Charleston, S. C. by Capt. Moore to serve till April 11, 1862, transferred to Regimental Band June 30, 1861, assigned extra duty learning music July 1, 1861, admitted General Hospital, Charlottesville, Va. October 13, Typhoid Fever, furloughed for 30 days November 5, readmitted December 20, Convalesent (Lung Disease) returned to duty December 21, discharged on Surgeon’s Certificate of disability December 30, Chronic Bronchitis, description on enlistment, 27 year old Music Teacher, 6’ 0”, fair complexion, blue eyes, light hair

* This company subsequently became (1st) Company G, 6th Regiment South Carolina Infantry

M267: Compiled Service Records of Confederate Soldiers Who Served in Organizations from the State of South Carolina