The last Hanna-Barbera Tom and Jerry short premiered in 1958.
That cartoon was Tot Watchers, released by Metro-Goldwyn-Mayer on June 25, 1958. It was the 114th and final theatrical short directed and produced by William Hanna and Joseph Barbera for the MGM series. The pair had created the enduring cat-and-mouse formula at MGM beginning with Puss Gets the Boot in 1940.
The wording can cause confusion because MGM’s cartoon studio closed in 1957, but several completed shorts were released afterward. Tot Watchers therefore belongs to the 1958 release year even though production activity had already wound down. Tom and Jerry returned to theatrical shorts in 1961 with Gene Deitch’s Switchin’ Kitten, which was made by a different production team in Czechoslovakia.
This is also not the final Tom and Jerry cartoon ever made. Joseph Barbera later returned for The Karate Guard, released in 2005; the question specifically identifies the Hanna-Barbera MGM run.