Congress used what is called a "Saxbe Fix" to resolve the situation.
Here is the section that Clinton's nomination would run afoul of:
No Senator or Representative shall, during the Time for which he was elected, be appointed to any civil Office under the Authority of the United States, which shall have been created, or the Emoluments whereof shall have been increased during such time; and no Person holding any Office under the United States, shall be a Member of either House during his Continuance in Office.From First Read:
[Article 1, Section 6, Clause 2]
The usual workaround is for Congress to lower the salary of the job back to what it was so that the nominee can take it without receiving the benefit of the pay increase that was approved while the nominee was in Congress. This maneuver, which has come to be known as "the Saxbe fix," addresses the clear intent of the Constitution, to prevent self-dealing.
But many legal scholars believe it does not cure the Constitutional problem, because the language of Article I is so clearly an absolute prohibition: No senator or representative, period.
"The content of the rule here is broader than its purpose,” said Professor Michael Stokes Paulsen, a Constitutional law expert at St. Thomas School of Law in Minneapolis. “And the rule is the rule; the purpose is not the rule.”
"A 'fix' can rescind the salary,” he added, “but it cannot repeal historical events. The emoluments of the office had been increased. The rule specified in the text still controls.”
I may not be a Constitutional law expert, but I would agree with Prof. Paulsen. Time cannot be turned back; the salary was raised. That cannot be undone. Therefore, I am of the opinion that Hillary Clinton is inelligible for Secretary of State.