I get some of my info from tombstones- both government and family stones that identify the unit fought in OR inscribed with GAR or UCV OR stones with various symbols (crossed guns, flags, etc.) The same from on-line cemetery lists.

I also compare cemetery records to the 1890 and 1910 censuses for Oklahoma. Also to the Territorial Assessors' Census Records for each county (so far only Logan, Payne and Pawnee have been found). These list veterans.

I also compare the cemetery lists to the Civil War Pension Index on ancestry.com. This usually only works when the wife survived the husband.

I also look for obits, funeral records that indicate pensioner or GAR or UCV paid for service.

Several other minor ways.