I agree with Steve that your GG Uncle was likely the man in the Mobile County Militia. There was no soldier named Jesse Reaves in the 10th Alabama Regiment. Company A of that regiment was raised in St. Clair County, and a man named Jesse Reeves joined that regiment on June 4, 1861, when your ancestor would have been about 15. On the 1860 census of St. Clair County, we find a Jesse Reeves living in the household of 62 year old Louisa Reeves. He is likely her son or grandson. He is 20 years old, and is far more likely to be the man in Company A, 10th Alabama Regiment. The 10th Alabama was never in South Alabama, having served the entire war in Virginia, Maryland, and Pennsylvania (Gettysburg). Pvt. Jesse Reeves was among those who were surrendered by General Lee at Appomattox Courthouse on April 9, 1865. Part of the confusion in your records has to do with the fact that the reference to Wilcox that you have found has nothing to do with the county of the same name, but is referring to the commander of the brigade to which the 10th Alabama belonged. See link below:

http://ehistory.osu.edu/uscw/features/regimental/alabama/confederate/wilcox.cfm