... or the emoluments whereof shall have been increased during such time ...
It is the pay increase (regardless of who authorized it) that disqualifies any sitting Senator or Representative from Cabinet (or other civil office) appointments.
The Nixon/Ford example wouldn't apply unless the pay (or other benefits) of the Office of the Vice President were increased. I don't know if they were ... but I'm betting they were not.
That would mean that Senators have the worst it, with 6-year terms to outlast. I guess the lesson is, if you aspire to be appointed to a Cabinet position, don't be in the Senate first.
An opposition outgoing President could take advantage of this in the future by raising the pay of Cabinet positions by $1, thereby disqualifying opposing party Senators for consideration by an incoming President for up to 6 years. A really mean President can wreak havoc, since Congressional terms begin on Jan 3, and Presidential terms begin on Jan 20. Theoretically, an outgoing President could use Executive Order to raise the pay of Cabinet members by $1 on Jan 4, making any Congressmen chosen by the incoming President-Elect suddenly disqualified for the position, since they aren't officially nominated or confirmed until after the new President takes office.
-PJ