To appoint a new Chief Justice, they would need to be confirmed by congress, perhaps nominated by the President? Not sure on the required legal/constitutional procedures. But that would require that the existing Chief Justice is no longer the Chief Justice. To be clear here, I am not implying any sort of violence against anyone, just that there must be some course of events that would lead to that scenario happening. Supreme Court judges, as far as I know, cannot just be replaced at whim. What I was trying to point out is that if the Oath of Office must be administered by the Chief Justice of the Supreme Court, then any inauguration would need to occur in the same location as the Chief Justice of the Supreme Court. What will actually happen is not something that is known to me, and I'm not making any predictions. Just putting out some thoughts in this strange time.