Actually discharges were quite common. A reason for discharge from service frequently appears in the service records. After May 1862 it could be that Confederate military law didn't require the man to be enrolled. The law of April 16, 1862, included so many exceptions that we can't review them all here. The age limits were 18 to 35 (the upper limit soon raised to 40 and eventually 45 by mid-1863), so that's one possibility. The law also allowed a man to produce a substitute not otherwise required to serve, and that's another.
Of course Confederate soldiers who enrolled in 1861 for a twelve-month term were discharged at the end of their enlistment.
Here are three other common reasons for discharge --
1) disability due to wounds or disease,
2) election or appointment to any civil office, which included county posts such as superintendent of education, commissioner or justice of the peace, one for every beat or precinct.
3) election or appointment of an enlisted man as a commissioned officer in another command. Transfer to the Confederate Navy or Marines also required that a man be released by means of discharge.
Do you know the reason for discharge?