Stacey Abrams’s defeat in the Georgia governor’s race was only a few weeks old when she arrived in New York in December to meet with campaign donors and political allies. At a reception in Manhattan’s Chelsea neighborhood, a supporter asked her what she would do next.

Ms. Abrams, attendees recalled, said she was undecided, except on one point: She was determined to seek high office again.

Since that meeting, Ms. Abrams’s next political moment has arrived with startling speed. She is set to give the Democratic response to President Trump’s State of the Union address on Tuesday night, a task of extraordinary prominence and an unheard-of distinction for a candidate who fell short of victory in the midterms.

But Ms. Abrams’s planned rebuttal to Mr. Trump is only one element of the role she is positioned to play in national politics: Democratic Party leaders are already imploring her to put her name back on the ballot, this time as a challenger to Senator David Perdue, a Georgia Republican who is loyally aligned with Mr. Trump. Democrats believe that by challenging Mr. Perdue in 2020, Ms. Abrams could help break the Republican Party’s near-monopoly on Southern power in the Senate, and perhaps help make Georgia competitive in the presidential race.