Rick,
I know you are looking for his Regular Confederate Service, If I remember Correctly he was also served in a 30 Day Volunteer Regiment, in 1861, before entering into Confederate Service in 1862.Researching his 30 day Volunteer Service, you may find out more information about him. Some of the 30 day Volunteer units were composed of Arkansas Militia Units, and research on the volunteers often show what militia unit and what confederate units the 30 day units went into.
I would also check to see if he was a Mexican War veteran. I know there was an Oliver Basham in the Mexican War, I don’t Know if it is the same man. Hope this helps.