Lawmakers in a badly divided House struggled mightily to agree, finally selecting Jefferson on their 36th ballot.
The excruciating muddle led to the adoption four years later of the 12th Amendment to the US Constitution, to somewhat clarify procedures around the election.
This time, if such a House vote should be necessary, it would take place on January 6, 2025.
How would that vote proceed?
One state, one vote
"Each state, regardless of population, casts a single vote for President in a contingent election," according to the Congressional Research Service (CRS).
In other words, Republican-leaning Wyoming with its city-sized population of 500,000 would have the same influence as Democratic California, where 39 million people live.
Though the US capital city Washington has three Electoral College votes, it would not get a vote in a contingent election, as it is not a state.
States with two or more representatives would need to hold an internal vote to determine which candidate to support, a CRS report said.
A candidate needs to win a majority of the 50 states or 26 votes. At present, that would likely give Republicans the edge.
Specific rules to govern the process would likely have to be adopted by the House, potentially leading to intense disagreements and a protracted period of constitutional crisis.
It is not hard to imagine how such a process, at the end of a razor-close campaign, would weigh on the already badly frayed nerves of American voters, many of whom are convinced that the vote has been rife with irregularities.