But Article I, Section 3 says more. In describing the powers of the Senate to conduct an impeachment trial, it provides that “Judgment in cases of impeachment shall not extend further than to removal from office, and disqualification to hold and enjoy any office of honor, trust or profit under the United States” (emphasis added).
That latter clause is the key, because it drives home that the Senate has two decisions to make in impeachment cases: First, it must decide whether an officer should be removed. Then it must decide whether this person should be disqualified from holding any future federal office. Indeed, of the eight officers the Senate has ever voted to remove, it subsequently voted to disqualify only three of them — reinforcing that removal and disqualification are separate inquiries. And as this procedure and historical practice make clear, by the time the Senate votes on disqualification, the officer has already been removed. In other words, disqualification, at least, is itself necessarily a vote about a former (as opposed to current) officer.
More than that, the disqualification power is both the primary evidence of and the central reason the Constitution allows for the impeachment of former officers.