We’re so used to the new Congress convening on January 3 (or nearly then, if it’s a Sunday) that it’s easy to forget that the date was, like for presidents and vice presidents, March 4 for most of the history of the Republic. There was typically a fairly long lame-duck session of the old Congress that opened in December, after the elections. The possibility that such a lame-duck Congress would decide the presidency, if it was thrown into the House of Representatives, was one of the things that the 20th amendment was designed to prevent.
The amendment also clarified what would happen if the president-elect happened to die between his election and his accession to office: “If, at the time fixed for the beginning of the term of the President, the President elect shall have died, the Vice President elect shall become President,” it says. The amendment addressed a circumstance that very nearly happened less than three weeks after its ratification, when an assassin killed Chicago Mayor Anton Cermak and injured other people while (presumably) attempting to kill President-elect Franklin Roosevelt.
Also, the 20th amendment specifies that if a president hasn’t been chosen by January 20, but a vice president has been, then the vice president-elect would act as president until things were sorted out.
Hard to imagine? If a third party actually got some traction in some future presidential election, and no set of presidential and vice presidential candidates received the need majority in the electoral college, then the two winners would be selected by different houses of Congress. The House chooses the president, the Senate the vice president. If the Senate were able to chose, but the House were deadlocked, then the Senate’s choice would act as president after January 20. It would be a peculiar thing, but possible.