I was actually thinking about this a couple days ago. Jefferson and Burr had won 73 Electoral Votes, Adams 65. You need exactly 70 to win. All Adams needs is 5 more vote somewhere. Because this was the pre-Jacksonian era, every state had some convoluted rules regarding how Electoral Votes were allocated.
An idea I had was just flipping New York. 12 Electoral Votes, all went to Jefferson and Burr, and they were chosen by the state legislature. The Federalists had lost the state legislative elections that year (and would lose the Governorship to George Clinton in 1801), but had been doing fairly well the last few years (and would even win back one house a few times before they dissolved permanently). Have the Federalists keep the state assembly, appoint 12 Adams electors, and Adams wins a second term.
Alternatively, Pennsylvania. In
1796 Jefferson had narrowly win the popular vote there (taking 14 of 15 Electoral Votes because people voted for individual electors instead of the candidates themselves). By 1800, the was no statewide Presidential election. Back then the legislature had to write the rules for each and every election, and the Federalists in the state legislature were afraid that Jefferson would win a statewide election, so they tried to gerrymander the votes (having districts elect one elector each). Eventually the state legislature appointed the electors (8 Federalist, 7 Jeffersonian). Either have the Federalists do a bit better in the 1799 elections and pull off their gerrymander plan, or have them do worse, have a statewide election of electors in 1800, and have Adams pull off a surprise win.
The Federalists weren't completely dead in 1800. An extended Adams Presidency could extend their life for the time being. Or maybe doom to it to an even earlier death.
Also, because on OTL Rhode Island elector voted for John Jay instead of Charles C. Pinckney, there would be no contingent election. Adams would be elected President, Pinckney Vice-President, which might possibly lead to the 12th Amendment being butterflied away or at least delayed.