I’m not sure that’s exactly correct. Sam Rayburn was Speaker of the House. At the time the Presidential Succession Act of 1886 would have been in place. From Wikipedia:
Had they been elected, the Presidential Succession Act of 1886 would have been invoked upon Willkie’s death, and his Secretary of State would have been sworn in as acting president for the remainder of the term ending on January 20, 1945.
Cordell Hull was Secretary of State at the time.
After Roosevelt’s death, the Presidential Succession Act of 1947 put the Speaker of the House as next in line to the presidency.