There have been very few Presidents who served in public office after the Presidency. JQA is by far the most famous of them, due not only to the mere length of his tenure in the house, but also due to the tenacious reputation that he earned while serving, where he became of the leading voices of abolition within Congress, fighting against the Gag Rule, the annexation of Texas, and participating in the famous legal case concerning the slave ship Amistad. He was the only former President to go to the House after, but not to Congress, joined in that by Andrew Johnson, although roughly opposite in terms of honor, Johnson serving a mere few months in the office in 1875 before dying in July of that year, and leaving no legacy to speak of.
There were a few others who had notable post-presidential political careers in highest level of government though. The next most obvious would be William Howard Taft, who was appointed as the Chief Justice of the Supreme Court by the next Republican, President Harding, in 1921, 9 years after he had failed to win reelection. It is often said that this had always been his true ambition anyways, and it is generally agreed that not only did he enjoy his time on the bench a great deal more than in the White House, but also that he was much better at it too, establishing himself as an able and forceful leader of the Court through the 1920s, generally seen as consistently conservative in how he kept the court directed, as well as being a strong advocate for legal reform from Congress, resulting in the creation of the Conference of Senior Circuit Judges and the passage of the Judges' Bill.
The final former President to highlight would be John Tyler, who I'm going to focus on here because conversely he is the most obscure on this count! After the Presidency he returned home to Virginia, and he did attempt to keep his reputation burnished, but didn't seek major office. With the decline of the Whig party, he began to turn toward the Democrats. In 1860, with deteriorating national situation, he attended the 1860 Democratic Convention, and although he didn't campaign for it, "entertained the delusion" that he might perhaps be offered up their Presidential candidate, under the impression that he would provide a unity candidate for the entire south to coalesce around. The odds of this were, in fact, nil, and he in the end became a supporter of Breckenridge, the more hardline candidate, announcing his impression of the situation to be "live or die, survive or perish."
Nevertheless, he wished not to see the Union perish, even if he feared it might be impossible to avoid, and as the wave of secession began, he was a supporter of the Crittenden Compromise, which had hoped to over a means to ensure slavery had stronger protections to alleviate concerns over Lincoln's election, but of course in the end failed. He continued to try to position himself as a force for compromise, offering to head a conference of the six slave and free states closest to the border, but this too didn't prevent the march to war, even if serving in the Virginia delegation gave him further chance to feel like he was trying. He saw a few other honors in the period though, being picked by Virginia to head a delegation to Pres. Buchanan to discuss the crisis, as well as to later meet with Lincoln on the eve of his inauguration.
His return to public service continued further with his selection to attend the Virginia convention for secession, where he was considered one of the most honored members, and by that point had shifted to being pro-secession, and by the vote on the 17th, had become one of the vocal proponents, and afterwards gave a public speech comparing their coming struggle to that of their revolutionary forefathers
With secession a done deal, he finally returned to public office, standing for election to the Confederate Congress, and being chosen by Charles City County to represent them in the House. Congress wouldn't meet until February, 1862, so he spent the intervening months negotiating the official terms for Virginia's entry into the Confederacy, as well as the agreement to move the capital of the wannabe nation to Richmond, from Montgomery. He traveled to Richmond to begin his new position... and died a month before the Congress convened, passing away on January 18th, 1862. In the United States, his passing received perhaps the least notice of any former president, the traitor's death going without comment from Lincoln and the government, while in Virginia, a 150-carriage funeral procession and great mourning accompanied his passing.
Sources
Burns, Kevin J. “Chief Justice as Chief Executive: Taft’s Judicial Statesmanship.” Journal of Supreme Court History 43, no. 1 (March 2018): 47–68
Crapol, Edward P.. John Tyler, the Accidental President. Chapel Hill: University of North Carolina Press, 2012.
Waldstreicher, David, ed. A Companion to John Adams and John Quincy Adams. Chichester: John Wiley & Sons, Incorporated, 2013.
ETA: Formatting and some clarity
As a a follow-up question, is there any precedent to a former president being considered for Supreme Court Justice?