Well, Kirkman then Speaker of the House Kimble, (then President pro Tempore of the Senate, they did appoint Senators earlier in the season, right?), then Moss.

This isn't the case. The only issue is that if Moss had been a two-term President (I don't think he was?), he would not be in the line of succession. He can still be SoS though — see Secretary Albright, who was not eligible for the Line of Succession due to not being a natural American citizen.