When Teddy Roosevelt retired from the presidency after 2 (arguably 1.5) terms, he certainly did not plan to remain a significant political figure. He had chosen his long time friend, William Howard Taft to succeed him, and probably assumed that it would be the end of his political career. Upon leaving the presidency, he embarked on a two year tour of Europe / African Safari.

His return from the trip was greeted with massive adulation, thousands came out to welcome him back in person. By this time Roosevelt also learned that Taft had embraced the establishment wing of the Republican Party, a group that TR had always kept at arms length. This, combined with a perceived feeling of malaise, let TR to challenge Taft in the primary (this is not a perfect answer to your question, because he did seek the office again). After loosing the primary (due to Tafts support among the Republican establishment) TR went on to found the progressive party, but would place 2nd (to Woodrow Wilson) in the election of 1908. Even after this defeat, he remained vocal, especially with the advent of WW1. He advocated for greater American involvement.

Source for TR: The Bully Pulpit by Dorris Kerns Goodwin

Final note: Some presidents have stayed in the limelight by holding other offices. John Quincy Adams was elected to congress, and Taft became a Supreme Court justice. (Those are the two I can recall at the moment, there are probably others)

Teddy Roosevelt remained quite active in the political arena. Taft served on the Supreme Court after being president. Bush helped campaign for his brother, and I do remember Clinton having a lot to say for a while after he left.

Though a precedent has mostly been set to stay out of it as a sign of respect.