There was no way Claire was going to just stay by Franks bedside while he recovered. She takes this as a golden opportunity to start putting herself in a power position. This is made much easier by the appointment of Donald Blythe as the Acting President. The way that Claire effortlessly manipulates Blythe is so enjoyable. Hes such an incompetent man that he doesnt even realize hes being turned into a puppet. And Claire takes him for everything hes got - allowing her to assist with negotiations with Russian President Petrov, attending the G7 Summit - and while she smiles and talks softly to Blithe, its clear the lack of respect she has for him. Its a nice little subtle moment that Claire refuses to share a cigarette with Blythe, something she often did with Frank during their scheming. Whatever issues she has with Frank, she knows Blythe is not on his level.