Born in New London, Missouri, Stout attended the common schools, Warrenburg (Missouri) State Normal School, and the University of Missouri. He taught school. He studied law. He was admitted to the Missouri bar in 1901 but did not practice. He moved to Lewistown, Montana, in 1902. He engaged in the newspaper business and was editor and publisher of the Fergus County Democrat 1902-1916 and the Lewistown Democrat News 1916-1946. He served as delegate to the Democratic National Convention in 1908 and to all State conventions from 1904 to 1946. He served as member of the State senate of Montana 1911-1913.
Stout was elected as a Democrat to the Sixty-third and Sixty-fourth Congresses (March 4, 1913 – March 3, 1917). He served as chairman of the Committee on Expenditures in the Department of the Interior (Sixty-fourth Congress). He was not a candidate for renomination in 1916. He served as member of the Montana Public Service commission 1930-1932.
Stout was elected to the State house of representatives in 1942, 1944, and 1946. Editorial writer for the Billings Gazette from 1947 to 1960. He married Sibyl Ann Sherlock, whose father, Henry Lloyd Sherlock, had been sheriff of Broadwater County, Montana, and a Montana state senator. She had two children from a previous marriage. Tom resided in Billings, Montana, until his death there on December 26, 1965. He was interred in Mountview Cemetery.