Senator Orrin Hatch, the oldest and most senior sitting Republican in the Senate, is not running for re-election. He’ll leave at the end of his term.
Batter up: Mitt Romney.
The most senior Senator of the party in control is normally the president pro tempore of the Congress and thus in the line of presidential succession. This is not a law. In fact, the Senate can elect a president pro tempore any time they want, and in the old days, typically elected a new one every time the Vice President was not presiding. This is one of those things that has always been done a certain way but need not be. The Senate can in theory elect any Senator of any party or any age at any time for this job.
I am a bit surprised that the Trump administration has not figured out that the Senate is following a tradition here, and interfered with it, just for fun. I assume they are ignorant of how this works.
Anyway, if they follow the pattern, the next president pro tempore might be Patrick Leahy, Thad Cochran, Juck Trasssley, Mitch McConnell or Richard Shelby. McCain is next in line, but he is leaving. Feinstein is after McCain, and she is being seriously challenged in the election.
There are many things to consider, including who among these very old people is alive, who is re-elected, and what party is in charge.
It is important because the individual will be third in line for the presidency. With Hatch retiring, I expect the whole thing about the president pro tempore will be discussed in the white house and Tump will insist that this spot be filled by Roy Moore or somebody.