According to Jon Michaels, a professor at the UCLA School of Law, author of Constitutional Coup: Privatization's Threat to the American Republic,
and renowned expert on constitutional law, presidential powers, public ethics, and conflicts of interest, that was not a requirement under the constitution at the time.
In actuality, it still isn't. A president's tenure, which the Constitution defines as lasting four years, must expire at noon on January 20 per the 20th Amendment's provisions.
The Constitution does not, however, specify how it should be handled. Instead, it's a question of custom.
The Electoral College was deadlocked when Thomas Jefferson waged a divisive political campaign against John Adams in 1800, and the House of Representatives had to declare the winner.
Adams still left his position in peace when the dispute was resolved, setting the standard for the following 220 years.