That is exactly what Obama did while Hillary was Secretary of State.
“Throughout the tenure of Secretary of State Hillary Clinton, there was no permanent inspector general at the State Department. Moreover, President Barack Obama did not nominate anyone for that position while Clinton was Secretary. Although the office remained running under Harold W. Geisel—a State Department employee—the “Government Accountability Office, lawmakers from both parties and nonprofit groups” criticized the delay in identifying a nominee, and called for the position to be filled by an inspector independent of the State Department.”
References:
Tau, Byron and Nicholas, Peter.
“”State Department Lacked Top Watchdog During Hillary Clinton Tenure”,
Wall Street Journal (March 24, 2015).
Smith, R. Jeffrey (April 25, 2011). “Vacant State Department inspector general position draws criticism”.
The Washington Post