The polite way for a President to leave the White House is to go to his successor’s inauguration. But there have been several presidents who have refused to attend their successor’s inaugurations.
The first President to avoid his successor was John Adams, after the rather nasty election of 1800:
The night before the inauguration of Thomas Jefferson, the lights burned late in the White House. President Adams’ belongings were packed into several wagons, and at four in the morning, President and Mrs. Adams boarded their coach and left the city. President Adams, taking his defeat personally, could not bring himself to attend the festivities attached to his defeat.
The next President was John Quincy Adams, after another bitter election loss, this time to Andrew Jackson:
During the days before the inauguration of Andrew Jackson, wagons were constantly shuttling between the White House and the mansion on Meridian Hill that Adams had rented after he and his wife decided to remain in Washington. During his lame-duck period (between the election and the inauguration of his successor), President Adams and his wife remained in the White House, trying not to notice the continuous celebrations being held to honor Jackson’s victory, and Adams’ defeat. After moving to his rented house on March 3rd, the day before the inauguration, they remained in their mansion. He did not attend or participate in any of the inaugural festivities.
The last one was Andrew Johnson, who was feuding with his successor, Ulysses Grant.