Here's a wikibox I copied from New Deal Coalition Retained: A Sixth Party System by
@The Congressman with the cabinet of President George Wallace in 1969 (The picture didn't go through so I had to type it out). Now most of these people I haven't heard of before so I'm not sure if this is the most plausible list for this timeline but I am sure some of these names would fit like Curtis LeMay as Secretary of Defense for example.
President: George Wallace
Vice President: Robert McNamara
Secretary of State: Richard Helms
Secretary of the Treasury: John Connally
Secretary of Defense: Curtis LeMay
Attorney General: Frank Lausche
Postmaster General: John A. Gronouski
Secretary of the Interior: John Melcher
Secretary of Agriculture: Robert Docking
Secretary of Commerce: Miles E. Goodwin Jr.
Secretary of Labor: Walter Reuther
Secretary of Health, Education, and Welfare: Daniel Patrick Moynihan
Secretary of Transportation: Sargent Shriver
White House Chief of Staff: John J. McKeithen
Spoilers ahead for those who have not read this timeline! For a reshresher of the timeline until 1968 Vice President Richard Nixon is elected President in 1960 with New York Governor Nelson Rockefeller as his Vice President. President Nixon is them assassinated by Lee Harvey Oswald in December 1963 leaving Vice President Rockefeller as President. President Rockefeller is defeated for re-election in 1964 by Massachusetts Senator John F. Kennedy (who lost the 1960 Democratic nomination to Texas Senate Majority Leader Lyndon B. Johnson) because of his divorce from his wife Mary Rockefeller and his subsequent remarriage to Happy Fitler. President Kennedy surprisingly lost re-nomination at the 1968 Democratic National Convention to former Alabama Governor George Wallace who proceeded to be elected President and serve two terms with President Kennedy's warhawkish Secretary of Defense Robert McNamara as his Vice President.