Born in Thibodaux, Lafourche Parish, Louisiana, Montet attended the common schools and Louisiana State Normal College at Natchitoches. He was graduated from the law department of Tulane University, New Orleans, Louisiana, in 1913. He was admitted to the bar the same year and commenced practice in Franklin, Louisiana. He served as secretary-treasurer of the city of Thibodaux in 1914 and as city attorney in 1915. He served as a member of the State house of representatives 1916-1920. He was an unsuccessful candidate for attorney general of Louisiana in 1924. He served as a delegate to the Democratic National Conventions in 1924 and 1932. Acting prosecuting attorney for the twentieth judicial district of Louisiana in 1925. He served as general counsel for State highway commission in 1928 and 1929.
Montet was elected as a Democrat to the Seventy-first Congress to fill the vacancy caused by the death of Whitmell P. Martin. In the special election, Montet defeated a Republican, M. E. Norman, 11,460 (57.7 percent) to 8,399 (42.3 percent). The Republicans finally won this House seat in 1972 but lost it again in 1980.
Montet was reelected to the Seventy-second, Seventy-third, and Seventy-fourth Congresses and served from August 6, 1929, to January 3, 1937. He was an unsuccessful candidate for renomination in 1936. He resumed the practice of law in Thibodaux, Louisiana, where he resided until his death there at the age of 93 on October 12, 1985. He was interred in Assumption Catholic Cemetery, Plattenville, Louisiana.