On the off chance that Senator John Cornyn is selected to be the next director of the FBI, Texas might just have the chance to fill his seat with a Democrat sooner rather than later.  From The Texas Tribune:

Gov. Greg Abbott would be tasked with a short-term appointment, but several months later the state would hold a special election to finish the duration of the term, which ends in 2021. 

The The Austin American-Statesman floats a couple possibilities as challengers to Gov. Abbott’s appointee: US Representative Beto O’Rourke of El Paso, who has already begun his 2018 campaign against Senator Ted Cruz, or US Representative Joaquin Castro of San Antonio.

That said, the scenario is unlikely to play out.  Again, from The Austin American-Statesman:

While Cornyn did serve as both a Texas Supreme Court justice and state attorney general before being elected to the Senate, he would seem among the least likely picks on the list.

Which leaves us here in the Lone State State waiting for the 2018 election for the next opportunity to elect a Democratic candidate to a Senate seat.