Nah. If you don't win, you don't win. The Constitution is silent on the issue of political parties, and what do you do when there are multiple losing opponents? Would you really have wanted Governor Thurmond to go into the Senate after losing in 1948?
And what about those who didn't quite run but received electoral votes from faithless elector (so, technically, were "candidates")? A freaking Senator-for-life Walter B. Jones in 1956?
In theory, it could be the candidate with the second highest number of electoral votes.
In such case, is a runner-up is an incumbent Senator, I assume he'd simply move from an elected to a lifetime seat.
There would be a vacancy in the senate seat, in that instance (as there were in 2009).
I really think that people come in second for a reason, and don't like the idea too much.