Eisenhower had demanded the inclusion in the cabinet of one woman, who turned out to be the Texas publisher Oveta Culp Hobby as secretary of the Health, Education, and Welfare Department soon to be created.
Oveta Culp Hobby should be better known.
The Wikipedia article on her says: “She was the first woman who was considered for a United States presidential candidacy by an incumbent United States President; Eisenhower encouraged her to run for president in 1960, but she did not run.” The article cites for this assertion:
Smith, Jean Edward, Eisenhower in War and Peace (N.Y.: Random House, 1st ed. 2012 (ISBN 978-1-4000-6693-3)), p. 756.