Tough enuf to research a common surname like Smith, then to find Carlton was really William Clayton!!
Here is what I found.Serving TN for the Confedarcy were 64 William Smith, no middle initial given.In addition, there were three privates named William C., one corporal named William C.And one private William C.O.Smith.
Almost none of these records give the person's residence, none give the age at enlistment, so figuring which might be the correct one will be most difficult.(about the only useful data would be the unit and company he served in).
Two ideas.You mention she has a photo of him.Are there any stripes on the upper arm of his uniform?If so, possibly he is the William C. who was the corporal.However, many of the William with no mid initial given were also corporals.The best chance is to see if a state archive has a record of a pension given for his service.The pension would be in the state he was living in when the pension was applied for.