The media blitz coincides with the publication of an illustrated children’s book she wrote, “Bold & Brave: Ten Heroes Who Won Women the Right to Vote.” It is set to be released Nov. 13.

Ms. Gillibrand has planned a small book tour, which includes stops in Washington, D.C., Northern Virginia, San Francisco, Los Angeles, Philadelphia, Albany and New York City through mid-December.

Ms. Gillibrand, who was appointed to the Senate in 2009 to replace Hillary Clinton after Mrs. Clinton was nominated as secretary of state, won roughly two-thirds of the vote in her election on Tuesday. The 3.73 million votes she received were the most for any candidate in New York this year — nearly 400,000 more than Gov. Andrew M. Cuomo, a fellow Democrat who is also considered a potential presidential candidate. He just won election to a third term as governor and has denied interest in a 2020 run.

Ms. Gillibrand spent minimally on her campaign. Her campaign chest actually grew from the beginning of the year through October, the opposite direction of most bank accounts during campaigns. She has more than $10.6 million in the bank, all of which could be used to jump-start a presidential run.

Unlike some possible 2020 contenders, like Senator Cory Booker, Democrat of New Jersey, who visited multiple key early primary states this fall, Ms. Gillibrand took only one trip to New Hampshire, where she campaigned for the Democratic candidate for governor in October.

The Democratic presidential primary is expected to be unusually crowded. In an interview during her New Hampshire trip, Ms. Gillibrand suggested that as many as eight Democratic senators could run for president in 2020, though she made clear she was not counting closely.

“I wouldn’t be surprised,” she said at the time. “People do not like President Trump and they are concerned and they would like to change that.”