Presidential Line Of Succession

Had the conspiracy been successful, who would have been the next president? What would be the process in determining/declaring the next president?

The Vice President-Elect gets sworn in if the President-Elect dies or becomes incapacitated, so during his first term it would have been Hannibal Hamlin - about whom I know nothing!

Not to split hairs, but if the VP elect hasn't been sworn in/isn't officially the VP yet, is there a "double" swearing in (take the oath as VP, then as President)?