Who were the 6 men that ran for President in the election FOLLOWING their their losing a Presidential election ? (Major parties, major players - Norman Thomas et al. do not count)

Thomas Jefferson (L 1796, W 1800) Democratic-Republican
Charles Pinckney (L 1800, L 1804, L 1808) Federalist
Andrew Jackson (L 1824, W 1828) Democratic-Republican
William Jennings Bryan (L 1896, L 2000) Democratic
Thomas E. Dewey (L 1944, L 1948) Republican
Adlai Stevenson (L 1952, L 1956) Democratic

An excellent question, to which I hope this answer is sufficient. I suspect that Ross Perot does not pass major party muster.

I see that I need to research a little better, Fuse. You have 2 that I don't. I missed Pickney and Dewey. That means, of course, that there were 8 men, not 6. You are missing two very obvious ones. (And, yes, Perot does not count.)

Not to be forgotten, Grover Cleveland, the sitting President, lost his bid for re-election in 1888 but won the office back in 1892.

A few more:

John Adams (L 1789, L 1792, W 1796) Federalist
John Jay (L 1796, L 1800; also L 1789) Federalist
Aaron Burr (L 1792, L 1796, L 1800) Democratic-Republican

Note that in early years, major parties may have had more than one candidate, with the runner-up (regardless of party) becoming VP.

Ronald Reagan, although not the Republican Party's official candidate in 1976, received one electoral vote and, of course, won the General Election in 1980.

I feel like I am still missing somebody!