In Iowa Sunday, Mr. Hickenlooper, whose campaign recently lost some of its top staffers, acknowledged that running for president involves “a bunch of skills that don’t come naturally to me,” but vowed to stay in the race. He has until mid-March to enter the Senate race.

Some Democrats have also urged Gov. Steve Bullock of Montana to run for Senate, hoping the popular governor could win an unlikely Democratic seat in a red state. He would also have until spring to make that decision.

But in Texas, the filing deadline to challenge Senator John Cornyn, another potentially vulnerable Republican incumbent, is Dec. 9 — nearly two months before any primary votes are cast. And although Democrats have already recruited M.J. Hegar, a veteran who won attention for her 2018 congressional campaign, both Beto O’Rourke and Julián Castro are better known in the state, and both have been pushed to run for the seat.

The remaining members of Congress in the 2020 race hail from states that have laws allowing them to pursue two offices at once. (New Jersey recently passed legislation permitting it, a bill know as “Cory’s Law.”) But just because some candidates can run for both offices doesn’t mean they should.

In Hawaii, State Senator Kai Kahele is seeking Representative Tulsi Gabbard’s seat, telling Vice News that she has a “tiger on her tail, and she’s going to be in trouble.” In Massachusetts, a second woman announced on Monday that she’d be making a primary run for Representative Seth Moulton’s seat. At least five other Democrats have left the door open to jumping into the race.

But don’t get too excited about the field shrinking quite yet. As Mr. Swalwell leaves, the California billionaire Tom Steyer plans to enter the race.

“I wish him well,” Mr. Swalwell said quickly, before dashing off to afternoon events in his district.