"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."
Clinton was in the Senate when a pay increase passed for Secretary of State. Thus, under this clause, she is not eligible for the office. This situation has come up in the past and Congress has simply passed a law reducing the salary to what it was before.
The LA Times has more on this issue here.