If you have his discharge papers they don't have his date of enlistment? If he was in SpW 1898 and out 1920 then he "retired" from the service and would have been Veterans Pension Benefits, medical care. He also would have been in during WWI. Have you read how to research service records at www.archives.gov (NARA Library)? Also Check Dept. of Veterans Affairs.
There is a Harry Stone DoB: 12/12/1877 DoD: 09/25/19481ST SGT US ARMY WORLD WAR I and wife Eileen N. Stone DoB: 09/04/1901 DoD: 08/22/1996both buried Section GSite 544 in the Ft. Rosecrans National Cemetery in San Diego, CA