I get James E(W) Gillum(Gillham), Enlisted May 1861, Cooper County Missouri, in 16th Regiment, Co. F, Parsons's Brigade (Price's) from his SOS card. That would I believe be a Missouri State Guard Unit at the time. In the NPS SOS system he is listed as being enlisted as a Private with the 16th Missouri Infantry, discharged aa a Corporal which matches the Missouri State card. He attended a reunion of ex-confederates in Sedalia in 1882.
Below is the unit history of the "regularized" 16th as after it was known in the regular CSA. The refernce to Caldwell matches. Obviously this is one of the units that had a core group of men that likely stated the same through several different unit designations. As I've said before I'm a people person and really don't do unit histories very well, so someone else may have to step in and fill out the history.
16th Regiment, Missouri Infantry
16th Infantry Regiment was formed by consolidating Caldwell's 7th and Jackson's 7th Missouri Infantry Regiment early in 1864. The unit was assigned to Parson's and S.P. Burns' Brigade, Trans-Mississippi Department, and was active in Louisiana and Arkansas. It sustained 14 casualties at Pleasant Hill and 25 at Jenkins' Ferry. In May, 1865, the regiment disbanded. Its field officers wre Colonel Levin M. Lewis, Lieutenant Colonel Pleasant W.H. Cumming, and Major Jesse P. Herrell.