For the vice presidency, there has not been a non-consecutive vice president to serve. But the reason there are more vice presidents than presidents, and only by 3, is because some vice presidents have served more than one president, some have been dropped and replaced by another, and then with the passage of the 25th Amendment, some have been appointed to fill a vacancy.
Only two vice presidents have served two different presidents. George Clinton was the first as Thomas Jefferson's second vice president, and first elected under the 12th Amendment, and while President Jefferson left office, Vice President Clinton was reelected to a second term with the newly elected James Madison. Clinton would also become the first vice president to die in office in 1812 as President Madison was seeking reelection that year. Interestingly, Clinton's replacement Elbridge Gerry would be the second vice president to die in office. The other vice president to serve two different presidents was John C. Calhoun. He was elected vice president with a majority of votes in 1824, unlike the presidential winner. Calhoun ran on 2 tickets in the same race so he was guaranteed to win enough votes to be the vice president. John Quincy Adams became president in 1825 and by the 1828 election the incumbent vice president was running against the administration. The ticket of Andrew Jackson and Calhoun won but the would become so antagonistic toward each other that Vice President Calhoun resigned, becoming the first to do so, and win election to the Senate to better combat President Jackson.
While Clinton and Calhoun are the only two vice presidents to stick around for more than one president, some vice presidents haven't even been asked back for another term. Vice President Burr was not on the first ticketed election in 1804, Clinton was the first vice president to replace another in a second term of a presidency. Since Calhoun resigned in 1832, President Jackson got a new vice president for the second term, Martin Van Buren. Jump to 1860 when Abraham Lincoln and Hannibal Hamlin won the general election with a plurality of the popular vote, but four years later President Lincoln would win 55% of the vote but without Vice President Hamlin. For his reelection, Lincoln ran on the Union Party ticket which represented not just the unity of the States and the Federal Government but the political factions within. So he chose a southern Democrat that remained loyal to the Union during the sectional crisis, a.k.a. the Civil War. That Democrat was Andrew Johnson of Tennessee. Ulysses S. Grant would get a new vice president for his second term. Franklin Roosevelt would have 3 different vice presidents over his four runs at the office. Gerald Ford was the last incumbent to drop the incumbent vice president for a new one. But FDR is the last to drop a vice president and win with a new one and Ford was the first to try since.
Ford is also a product of the 25th Amendment. After the assassination of President John F. Kennedy in 1963, the threat of nuclear war brought the presidential line of succession into focus. Throughout the history of the presidency the vice presidency never had a remedy for a vacancy. Whenever the office was vacant it wasn't replaced until the next election and for some that was a whole four years. Such a thing could not happen in the Cold War era, so by 1967 the amendment was ratified. Just six years later in 1973, the amendment would be invoked and Gerald Ford would be the first appointed vice president. Then as President Ford, he too would appoint his own vice president, the moderate Republican Nelson Rockefeller...before dumping him in '76 for the more conservative Bob Dole.