Find him in 1930 census in Tacoma, Pierce Co WA where cemetery is located. Living in Son-in-law's household with daughter b. abt 1896 in Idaho, has Veteran-SP.
Spanish American War Pensions are included in the database "Civil War Pension Index...." on Ancestry, from the NARA source:"General Index to Pension Files, 1861-1934. Washington, D.C.: National Archives and Records Administration. T288, 546 rolls.":
a Peter J. Flynn filed in Idaho May 4th 1903 and has service as E, 2 & G, U.S. Cav. No widow listed.
"U.S. Army Register of Enlistments...." has Peter J. Flynn, enlisted May 12th 1897 in Walla Walla, WA, age 33 b. mayo Ireland assigned 4th Cav Company Glast service looks like 3 or E ls 4 Cav 5/11/1897 Discharged Feb 25th 1899SO#7 Also 99 Manila, P.I.Pvt "Very Good"
1900 census has Peter J Flynn with wife Edna, and 3 daughters including oldest Edna living in Wardner, Shoshone County ID.