The Constitution provides that the President “shall be removed from office on impeachment for, and conviction of, treason, bribery, or other high crimes and misdemeanors,” but it says nothing about the timing of when the impeachment and trial may take place. That omission makes sense, since presidents – and any other impeachable officials – could commit impeachable offenses at any time while they are in office, including in their last months or days in their positions. It certainly makes no sense for presidents who commit misconduct late in their terms, or perhaps not discovered until late in their terms, to be immune from the one process the Constitution allows for barring them from serving in any other federal office or from receiving any federal pensions. What’s more, litigation or prosecutions might not be able to get at the misconduct, since the scope of impeachable offenses extends to misconduct that is not an actual crime. And what if that misconduct is not discovered until after a president leaves office? There may be no practical means for holding him accountable for such misconduct, especially if he is regarded as having been immune from any criminal prosecution or inquiry while he was in office. Being president is not a safe harbor from political and legal accountability. This is why John Quincy Adams proclaimed on the floor of the House that, “I hold myself, so long as I have the breath of life in my body, amenable to impeachment by this House for everything I did during the time I held any public office.” (Michael J. Gerhardt, The Federal Impeachment Process: A Constitutional and Historical Analyses 80 (2d edition 2000) (citation omitted)) Adams’s suggestion was that any impeachable official remained subject to that process well after they left office, not just presidents but those who abused power while in office.
In 1876, Secretary of War William Belknap resigned as the House was considering impeaching him for a newly revealed corruption scandal. The House impeached him anyway, and the Senate rejected a motion to dismiss the case for want of jurisdiction over a former federal officer. Belknap was not convicted, in part because some senators doubted their authority to do so. Condemning Belknap's actions and disqualifying him from future office seemed a sufficient reason to proceed for many in the House and Senate.
If Congress in 1974 had imagined the possibility of President Richard Nixon rehabilitating his reputation sufficiently to have a chance at holding a future office, it is not hard to imagine a bipartisan House and Senate steaming ahead with an impeachment and trial in order to bar that possibility through a judgment of disqualification. Worried that an infamous former officeholder might eventually live down his infamy, Congress might seek to make that recovery more difficult through the impeachment process.
The House practice manual accepts that the impeachment power extends to former officers, though it admits that since removal is generally the "primary objective" of an impeachment the proceedings have usually been brought to an end if the officer resigns. Brian Kalt has provided the most comprehensive analysis of "late impeachments," and I find him persuasive.