Have any Vice Presidents later run for Governor or other office besides President after their terms?
Yes. Not counting those who served as President or ran for President following their time as Vice President, or the seven Vice Presidents who died in office (George Clinton, Elbridge Gerry, William R. King, Henry Wilson, Thomas A. Hendricks, Garret Hobart, and James S. Sherman), here are the VPs who sought other offices post-Vice Presidency:
•Aaron Burr (1801-1805): Lost race for Governor of New York in 1804 during his Vice Presidency.
•Daniel D. Tompkins (1817-1825): Lost race for Governor of New York in 1820 during his Vice Presidency.
•John C. Calhoun (1825-1832): Resigned the Vice Presidency to join in the U.S. Senate from South Carolina (1832-1843); Served as Secretary of State (1844-1845) in the last stretch of the Tyler Administration; Elected again to the U.S. Senate from South Carolina (1845-1850) after serving as Secretary of State.
•Richard M. Johnson (1837-1841): Lost race for the U.S. Senate in Kentucky in 1842; Served two separate terms in the Kentucky House of Representatives (1841-1843; 1850) after his Vice Presidency. Died two weeks into his second post-Vice Presidential term in the state legislature.
•John Tyler (1841): After serving as Vice President and President, and following Virginia's secession from the Union in 1861, Tyler was elected as a delegate to the Provisional Confederate Congress. Tyler was also elected to a full term in the Confederate House of Representatives but died just before taking his seat in February 1862.
•George M. Dallas (1845-1849): Appointed U.S. Minister to Great Britain (1856-1861) by President Pierce and served under Presidents Pierce and Buchanan before being replaced early in the Lincoln Administration.
•John C. Breckinridge (1857-1861): Elected to a U.S. Senate seat from Kentucky while still Vice President. After administering the oath of office to his successor as Vice President, Hannibal Hamlin, Breckinridge was immediately sworn into the Senate by Hamlin. Although Kentucky remained neutral during the Civil War, Breckinridge supported the Confederacy and joined the Confederate military while still a sitting Senator, resulting in treason charges in November 1861 and, a month later, unanimous expulsion from the Senate. Breckinridge became a general in the Confederate Army and served as Confederate President Jefferson Davis's final Secretary of War.
•Hannibal Hamlin (1861-1865): Briefly served as Collector of the Port of Boston (1865-1866) after being appointed by President Andrew Johnson. Elected U.S. Senator from Maine (1869-1881). Served as U.S. Ambassador to Spain (1881-1882) under Presidents Garfield and Arthur.
•Andrew Johnson (1865): After his brief Vice Presidency and nearly four years as President, Johnson lost races for the U.S. Senate (1869) and U.S. House of Representatives in Tennessee. Elected as U.S. Senator from Tennessee in 1875 and died in office.
•William A. Wheeler (1877-1881): Wheeler was considered as a candidate for the U.S. Senate from New York on several occasions following his Vice Presidency but never made a serious bid for election.
•Levi P. Morton (1889-1893): Served as Governor of New York (1895-1896).
•Adlai E. Stevenson (1893-1897): Lost race for Governor of Illinois in 1908.
•Charles W. Fairbanks (1905-1909): Fairbanks was the Republican nominee for Vice President on a ticket alongside Presidential nominee Charles Evans Hughes in 1916 but they lost to incumbent President Woodrow Wilson and Vice President Thomas R. Marshall.
•Charles G. Dawes (1925-1929): Served as U.S. Ambassador to Great Britain (1929-1931), appointed by President Hoover.
•Henry A. Wallace (1941-1945): After being dumped as Vice President in favor of Harry Truman when Franklin D. Roosevelt ran for a fourth term in 1944, FDR appointed Wallace Secretary of Commerce where he served from 1945-1946 under Roosevelt and Truman
•Alben W. Barkley (1949-1953): Elected to the U.S. Senate from Kentucky after his Vice Presidency and served from 1955 until dying in office in 1956.
•Richard Nixon (1953-1961): After losing his first bid for the White House in 1960, Nixon also lost a race for Governor of California in 1962 after leaving the Vice Presidency before making a remarkable comeback to win the Presidency in 1968.
•Hubert H. Humphrey (1965-1969): Elected to his former seat in the U.S. Senate from Minnesota and served until dying in office (1971-1978).
•Walter Mondale (1977-1981): U.S. Ambassador to Japan (1993-1996), appointed by President Clinton. In 2002, Mondale lost a race for U.S. Senate from Minnesota when he was the last-minute replacement on the ballot after Senator Paul Wellstone was killed in a plane crash.