The 1900 Census apparently shows Walter Lee Bailey was adopted by another family - the Palmers. It looks like by 1900, he was in his adopted family:
"United States Census, 1900," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:M3P3-ZVT : accessed 3 August 2021), Walter Palmer in household of Jo Palmer, Justice Precincts 2-3, Jones, Texas, United States; citing enumeration district (ED) 100, sheet 5A, family 90, NARA microfilm publication T623 (Washington, D.C.: National Archives and Records Administration, 1972.); FHL microfilm 1,241,650.
He kept that same name "Palmer" in the 1910 Census:
"United States Census, 1910," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MVV7-QZP : accessed 3 August 2021), Walter Lee Palmer in household of Ethen A Rose, Courtland, Cochise, Arizona, United States; citing enumeration district (ED) ED 14, sheet 16A, family 447, NARA microfilm publication T624 (Washington D.C.: National Archives and Records Administration, 1982), roll 38; FHL microfilm 1,374,051.
Then went back to "Bailey" when he got married:
"Arizona, County Marriages, 1871-1964," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:QG1X-TGV4 : 29 November 2018), Walter Lee Bailey and Elese Gordon, 1914; citing Marriage, Cochise, Cochise, Arizona Territory, United States, Arizona Department of Libraries, Archives, and Public Records, Phoenix; FHL microfilm 007541661.