The inspector general, Steve A. Linick, who leads hundreds of employees in investigating fraud and waste at the State Department, had begun an inquiry into Mr. Pompeo’s possible misuse of a political appointee to perform personal tasks for him and his wife, according to Democratic aides. That included walking the dog, picking up dry-cleaning and making restaurant reservations, one said — an echo of the whistle-blower complaint from last year. The details of Mr. Linick’s investigation are not clear, and it may be unrelated to the previous allegations.