Senate candidate would have a running mate much as presidential candidates do now. These vice senators would not be duty free placeholders like many vice presidents, but would simply be the person designated to replace the senator should they vacate their office for whatever reason. The vice senator would likely have to be on staff with the senator or in some other public office since remaining in a private sector job might put that person in a conflict of interest situation with the senator.
This plan would avoid situations where a Republican governor gets to choose a replacement for a Democratic senator and vise versa. Or situations where a governor gets to play politics or goes totally rogue and tries to sell the seat. It also eliminates the need for a costly special election since the vice senator would have been elected already as part of the senator's team.
Now, some my ask where all this ends. Should we have vice representatives too? Well, no. I think drawing the line with senators makes sense. They have six year terms. Reps have two year terms. You are never far from an election for reps. Additionally, any given representative is a much smaller player than a senator. I don't think we need to go as far as having vice representatives.
Solving problems one at time here at TEH.