The reason for such an impeachment would be then, after conviction, to disqualify that person from holding future office. Disqualification is a sanction that can be attached following impeachment and conviction.
The most prominent example of Congress having asserted such a power occurred during Reconstruction, when Secretary of War William Belknap resigned just before the House voted on the Articles of Impeachment. The House went ahead and approved the articles, and Belknap was then tried in the Senate a month later, which failed to convict him. Belknap attended his trial in the Senate. For the Senate’s history of this event, see here.
The leading academic article on the general issue, as far as I know, is Brian Kalt’s article, The Constitutional Case for the Impeachability of Former Federal Officials: An Analysis of the Law, History, and Practice of Late Impeachment. Here is his abstract, which summarizes the general history and arguments:
This article considers the constitutional case for the impeachability of federal officers after they have left office. As a practical matter, while it may rarely be worthwhile to pursue a late impeachment (as with regular impeachment), this does not change the fact that it can be done, or that certain facts may make it desirable. The article principally argues that: (1) Late impeachment was practiced in England and, unlike other aspects of English impeachment, was never explicitly ruled out in America. Indeed, some state constitutions made late impeachability explicit, or even required. (2) Structurally, impeachment is designed not just to remove but to deter, and this effect would be severely undermined if it faded away near the end of a term. Convicted impeachees can be disqualified from future federal office, an important punishment that should not be automatically mooted if the officer resigns or the president removes him. (3) The precedents are mixed, but the Senate has approved late impeachment. Senate opponents of late impeachment have not prevented late trials, and they cannot alter the formal declaration of a majority of the Senate in one case that officers can indeed be impeached after they have left office