Clearly presidents that were assassinated/died in office didn’t go to the inauguration of the next president. But has a one-term president ever not gone to the inauguration of the guy that beat them?
John Adams.
The election of 1800 was nasty in a lot of ways. Adams already wasn’t popular due to his administration’s support for the alien and sedition acts, which political opponents criticized as just another form of monarchy. There were other policy-related reasons that he was criticized, but what probably made Adams so distraught that he refused to attend the inauguration were the smear campaigns enacted by political operatives on both sides.
Jefferson and Adams had previously been friends, despite their differing political beliefs. But now ensconced in opposing political parties, the campaign attacks became increasingly personal and exaggerated. Adams supporters claimed that a Jefferson presidency would bring in an era of open murder, rape, and incest. Jefferson responded by hiring James Callender, a journalist who had been arrested over violating the controversial sedition acts, to write negative propaganda. Callender accused Adams of being a hermaphrodite (keep in mind that Adams is the guy who signed a law punishing negative speech about the president).
*An interesting footnote to this story: Callender was expecting to be rewarded for his service by Jefferson appointing him to a government job once he was elected president. When Jefferson refused, Callender turned on Jefferson and published rumors about him, including the now well-known one (generally accepted as being true) that Jefferson had fathered children with his slave Sally Hemings.
After Callender’s hermaphrodite rumors spread, Adams deliberately spread a rumor that Jefferson had died and thus was not an eligible candidate. Newspapers eventually rectified this, and in the end it didn’t seem to matter as Jefferson got over 60% of the vote and Adams got under 40%. (Although there were some reports of election irregularities and miscounting, which also may have contributed to Adams’ bitterness).
Just days before the end of the election, Adams’ son Charles died due to complications from alcoholism. When the inauguration came around in March Adams was still bitter over a humiliating loss, the death of his son, election irregularities, personal insults hurled at him during the campaign, and what he perceived to be the betrayal of an old friend. He left the White House before the sun rose on the day of the inauguration.