Lynn - I can find no evidence of this man in a Mississippi unit. I researched all of the James M. Lees, as well as three more who did not have a middle initial listed:

James M. Lee - age 56 - Enlisted in Monroe County - 28th Mississippi Cavalry
James M. Lee - age 30 - Enlisted at College Hill near Oxford - Hughes Battalion, Mississippi Cavalry
James M. Lee - age 25 - Enlisted at Molino in Union County - 23rd Regiment, Mississippi Infantry

James Lee - Enlisted in Simpson County - 1st Battalion State Troops, Mississippi Infantry (12 Months 1862-63)
James Lee - Enlisted in Wayne County - 17th Battalion, Mississippi Cavalry (only one card in CMSR - listed as Deserter)
James Lee - Paroled at Jackson following the surrender at Citronelle, Alabama, residence shown as Rankin County - 1st Regiment, Mississippi Cavalry