alternatehistory.com

Not particularly plausible, but I've already gone this far.

The Curse Of Ike

1953-1955: Dwight Eisenhower/Richard Nixon
1955-1957: Richard Nixon/Vacant
1957-1960: Lyndon Johnson/John F. Kennedy
1960-1961: John F. Kennedy/Vacant
1961-1962: John F. Kennedy/Estes Kefauver
1962-1963: Estes Kefauver/Vacant
1963-1963: Sam Rayburn/Vacant
1963-1963: Carl Hayden/Vacant
1963-1964: Adlai Stevenson/Vacant
1964-1965: Averell Harriman/Vacant
1965-1967: Richard Nixon/John Tower
1967-1967: John Tower/Nelson Rockefeller
1967-1968: Nelson Rockefeller/James Rhodes
1968-1969: James Rhodes/Margaret Chase Smith
1969-1971: Hubert Humphrey/Scoop Jackson
1971-1973: Scoop Jackson/Ted Kennedy
1973-1973: Ted Kennedy/Barbara Jordan
1973-1974: Ted Kennedy/Barbara Jordan
1974-1975: Barbara Jordan/Frank Church
1975-1976: Frank Church/Walter Mondale
1976-1977: Walter Mondale/Jesse Unruh
1977-1978: John Wayne/Dewey Jackson Short
1978-1979: Dewey Jackson Short/Milton Young
1979-1980: Milton Young/Gerald Ford
1980-1981: Gerald Ford/Alan Simpson
1981-1982: Jesse Unruh/Abe Fortas
1982-1982: Abe Fortas/Nick Begich
1982-1984: Nick Begich/Jimmy Carter
1984-1985: Jimmy Carter/Mickey Leland
1985-1987: Ronald Reagan/H. John Heinz
1987-1988: H. John Heinz/Lee Atwater
1988-1989: Lee Atwater/Dick Cheney
1989-1989: Lee Atwater/Dick Cheney
1989-1990: Dick Cheney/Spiro Agnew
1990-1991: Spiro Agnew/Bill Emerson
1991-1992: Bill Emerson/George Bush
1992-1993: George Bush/Bob Dole