What is the history of the tradition of the out-going president attending the inauguration of his successor?

I started looking into answering, but then ran across this summary of the three most noteworthy cases. I'm going to link it here as it does a pretty good job of the three cases that are applicable to the 2020 case.

The article is not 100% correct in that it lists three outgoing presidents, when there are actually five cases, though the restriction is reasonable. The five cases are:

• John Adams did not attend Jefferson's inaugural
• John Quincy Adams did not attend Jackson's inaugural
• Andrew Johnson did not attend Grant's inaugural
• Woodrow Wilson did not attend Harding's inaugural
• Richard Nixon did not attend Ford's inaugural

The first three cases can be put down to sour grapes or anger on the part of the outgoing president.

The race between John Adams and Thomas Jefferson was quite acrimonious, to the extent that it formed a breach in a friendship between the two men that would only return years later. Despite being probably only the second "real" election (i.e. one that wasn't a foregone conclusion), the shenanigans that followed it was not unlike what we've seen in 2020:

Jefferson clearly defeated John Adams in the fall of 1800, but he tied with his running mate Aaron Burr at 73 votes each in the cumbersome and imperfect Electoral College. That sent the election into the House of Representatives where lame-duck, bitter, sour grape Federalists attempted to unseat Jefferson by working out a political deal with the opportunistic Burr. Since the Constitution could not tell the difference between the presidential candidate (Jefferson) and the vice presidential candidate (Burr), the Federalists determined to exploit the technical ambiguity by entering into a corrupt bargain with Burr. This created America’s first constitutional crisis and though it is hard for us to believe 200 years later, the United States came close to civil war over the results of the election.

On Feb. 17, 1801, the Federalists in the House of Representatives finally, on the 36th ballot, gave up their quest to steal the presidency from Jefferson and certified his election. This left President-elect Jefferson just 15 days to put his administration together. It seems certain that he had already drafted his inaugural address — one of the three or four greatest in American history — and he knew going in that his principal adviser would be James Madison.

Some one imagines the bad-blood was still fairly hot. Also, we should give Adams a bit of a pass here because this was only the second transition of power, and the first one had not involved a transition to a new party. There was no solid tradition.

The second case involved John Quincy Adams and Andrew Jackson. That election was also famously fraught, with Jackson's win seen as revenge for the "corrupt bargain" that saw Adams win despite losing the popular vote. The following campaign that saw Jackson prevail was also an unusually dirty campaign.

Thus primed, both sides waged an incredibly dirty campaign in 1828. Adams was portrayed as extravagant and corrupt; Jackson was denounced as an American Caesar. Worst of all, the dubious circumstances of Jackson’s marriage were widely broadcast. His legendary temper was depicted as his defining characteristic. He had fought duels, killing prominent Nashville attorney Charles Dickinson in one for insulting his wife. He had brawled in the streets of Nashville, had threatened to cut off senators’ ears, and had executed militiamen under his command. The Adams camp hoped these stories would both persuade the people that Jackson was unsuitable and provoke him to additional outbursts that would bolster the impression

This bad blood ran up to inauguration day.

Most telling was Jackson’s treatment of John Quincy Adams. Although Jackson especially held Clay responsible for the ugly press attacks on Rachel during the campaign, he did not regard Adams as blameless. Moreover, Jackson was convinced that Rachel’s discovery of these reports had contributed to her death. Understandably bitter, he refused to pay a courtesy call on Adams during the three weeks before the inauguration. Adams rightly saw Jackson’s behavior as a deliberate snub and refused to attend the inauguration, something of an Adams tradition: his father had similarly slighted Jefferson after the rancorous campaign of 1800.

Again, remember this is a fairly early election, only the third that involved the winner not being the sitting president's pick. So the tradition was perhaps not solid.

The third involved the Civil War, and all the acrimony therein. As most should know, Andrew Johnson was not a popular president. Johnson brought Grant into his administration as a political move.

In August, Johnson struck at Secretary of War Edwin M. Stanton, who had long been a Radical agent in the presidential camp and was protected by congressional allies through the Tenure of Office Act, which prohibited removal of cabinet officers without the consent of the Senate. Johnson suspended Stanton and appointed Grant acting secretary of war. Johnson knew that he could not succeed in his high-handed removal of Stanton without replacing him with the most popular man in the country; Grant accepted rather than allow the army to fall into unfriendly hands.

But Grant's politics were not much in alignment with Johnson's, and he had a track record of butting heads with commanders, so it's no shock that it happened here:

The conflict remained quiet because Grant, as a soldier, was determined to obey the commander in chief and because Johnson needed Grant's popularity to shore up his political power. Johnson dragged Grant along on a "swing around the circle," a trip ostensibly to dedicate the Douglas tomb in Chicago but really a political tour to allow Johnson to argue before the voters his case against congressional Radicals, who demanded sweeping political and social change in the South. Johnson's undignified harangues disgusted Grant, who temporarily left the party at Cleveland, leading staunch supporters of Johnson to charge that Grant had withdrawn to recover from excessive drinking. Recognizing the dangers of their eroding relationship, Johnson tried to send Grant on a mission to Mexico and to bring William T. Sherman to Washington in his place; Grant flatly refused to go, insisting that the president had no authority to order an officer on a civilian mission.

When congress returned, it tried to undo Johnson's appointment, leading to direct conflict with Grant

Johnson and Grant managed this uneasy partnership until Congress reassembled at the end of 1867, quickly evincing a determination to reinstate Stanton and placing Grant in the untenable position of obeying either his commander in chief or Congress. Grant told Johnson that he intended to resign the office of secretary of war because to hold firm would make him liable to fine and imprisonment under the Tenure of Office Act. Johnson asked Grant to delay his resignation and believed that he had agreed to do so. Through misunderstanding (as Grant's friends believed) or bad faith (as Johnson believed), Grant surrendered the office to Stanton before Johnson had an opportunity to nominate an alternative candidate who might have garnered enough Republican support to achieve confirmation. The restoration of Stanton led to a stormy cabinet confrontation during which Johnson accused Grant of lying. Publication of the exchange of acrimonious correspondence that followed the cabinet meeting completed the process of rupture between president and general

Note that this is the incident that led directly to Johnson's impeachment. At the inauguration, Grant refused to share a carriage with Johnson (as is also traditional to this day) and Johnson retaliated by not attending the inauguration.

The CNN article does not mention the two more recent cases, but both are more mundane and clearly don't involve sour grapes.

Woodrow Wilson had a stroke near the end of his presidency, which likely left him unfit to serve.

Not knowing Wilson’s condition or prognosis, the cabinet—and the entire nation—spent the next 17 months paddling in a sea of hearsay, whispers, and speculation.

Only Grayson and, more importantly, Edith Bolling Galt Wilson, the president’s second wife, were regularly in the ailing Woodrow Wilson’s company and privy to his true condition, but neither was forthcoming.

For a year and a half, the United States of America operated under an unelected shadow government of two.

Wilson and his successor, Harding, did not have an acrimonious relation, and in fact Wilson participated in some inauguration activities and in fact were the first to do so in an automobile rather than a carriage:

President Wilson and President-elect Harding stepped into their automobile (for the first time in history), left the White House and headed down Pennsylvania Avenue toward the Capitol.

Wilson not attending the inaugural itself was more of a health issue.

His doctors and family recommended that he not go to any inaugural ceremonies other than to accompany the Hardings from the White House to the Capitol. The Salisbury(Md.) Evening Post reported that President Wilson was “walking feebly with the assistance of a cane” and that it was “necessary for secret service men to place his feet on each succeeding step as he descended as it was apparent to all that it would be impossible for him to take part in the ceremonies at the Capitol.”

The final case is Richard Nixon, and it should be obvious there both that there was no acrimony with the incoming president. In this case, it was not a ceremony planed in advance, but a quick response to Nixon's resignation. You could argue that it doesn't entirely count as it was more akin to an emergency oath taking ala LBJ, Teddy Roosevelt, etc. than a normal transfer of power.

On August 9, 1974, Gerald R. Ford became President of the United States and immediately set to work. He did not have the luxury of an inauguration ceremony as the country was in conflict, domestically and internationally. Gerald R. Ford’s first day as President and his focus on a nation in crisis are detailed through photographs which include his remarks following being sworn in as President, White House briefings and meetings with International Leaders.

So the summary: Five living presidents did not attend their successor's inauguration. Three because of sour grapes, one because of health and one because it wasn't a "real" inauguration ceremony.