On Capitol Hill, she has made addressing the epidemic of rape in the military a central pursuit, including sparring with military brass in congressional hearings. She has pushed legislation to address campus sexual assault, as well. In November, she became one of the lead co-sponsors of the “Me Too Congress Act” to improve the process for whistle-blowers and victims inside Congress.

Politically, she has created a political action committee to raise money and encourage women to run for office. She calls the PAC Off the Sidelines, the same name she gave her book. (Mrs. Clinton wrote the foreword.)

“I’m not surprised at all that she wants to be on the leading edge of these discussions,” said Josh Holmes, a Republican strategist and former chief of staff to the Senate majority leader, Mitch McConnell. “It certainly follows what her interests have been in the Senate.”

While some people have embraced her stances on Mr. Franken and Mr. Clinton, others have questioned the timing, especially on Mr. Clinton, after he and his wife have exited the political stage.

Matt Canter, a former aide to Ms. Gillibrand who remains in touch with the senator, said, “Many people are proud that, at a time like this, she is providing moral clarity both among people in public office and within the Democratic Party.

But Mr. Holmes noted: “It’s one thing to be critical to Bill Clinton. It’s another thing to be critical after his relevance is done.”