The NARA has Spanish American War Pension Cards on the microfilm Titled "U.S., Civil War Pension Index: General Index to Pension Files, 1861-1934"
Ancestry has images of this microfilm and Thomas H McEldowney with widow Erma has Thomas H on second line as a "alias" and the name his records maybe under is Herschel McEldowney. Erma filed in Ohio and the units written on card "Cas Detch. & C&H 29th US Infantry.
As Herschel McEldowney Ancestry has databases "U.S. Returns from Regular Army Infantry", "U.S. Army, Register of Enlistments, 1798-1914" (enlisted PA) Nand "U.S., Returns from Military Posts, 1806-1916" (has middle initial G returned 1904 to Military Post: USA General Hospital, Presidio, San Francisco, CA, he was also discharged from the Army at the Presidio)
A family tree online has he died in Salem, Columbania County Ohio.
1910 Census Thos and Erma are "Roomers" living in Youngstown. Mahoning County OH. Ancestry has a space between MC and Eldowney for 1910 census search so search McEldowney and Mc Eldowney to find all records, at least with Ancestry.
Look in Columbiana and Mahoning County for burial.