Well, I tried to find a Civil War military service online record for John Wood or John Woods (as well as his brother George) without success in the Missouri Sec'y of State's website under MO State Archives. There were lots and lots of men by those names to go through, but the information you provided (mainly that he lived in Vernon County) helped me from having to examine most of the records. Knowing that Woods or Wood lived in Vernon County helped me to eliminate all the local militia records except those close to Vernon County. Those records in Jefferson City are not complete, but it is discouraging. I did examine all "regular" units with men of those names in both Union and Confederate armies from Missouri.
I toyed with the concept that John was not a soldier at all, but a teamster employed by the US Army Quartermaster Corps, since the 1890 Veteran Schedule gave his rank as "Train" or "Team." However, that 1890 record clearly states that he "enlisted Fall 1862" and was discharged April 1865. That spells "soldier" to me, and teamsters were hired wage earners, so I eliminated that possibility.
If you haven't already, I strongly urge you to contact the Newton County Museum Library at 121 North Washington St. Neosho, MO 64850, which is also known as the Newton County Historical Society, and seek their help on this matter from local records.
In my humble opinion, without knowing more about Woods' service, you don't even have enough detail to inquire further in state or federal records.
I'm sorry that I could not help you in this matter.