Foreign policy was a signature issue of her late brother, Lindsey Graham.
OAP said McConnell is "not yet medically cleared to leave the rehab facility."
Former state lawmaker and logger Troy Jackson secured the Democratic nomination for U.S. Senate in Maine on Saturday.
Her announcement comes after President Trump endorsed her for a full term.
Illinois' two U.S. senators released statements reacting to the sudden death of South Carolina Senator Lindsey Graham. Sen. Dick Durbin called him "a fierce Republican partisan one day and a key bipartisan ally the next."