In addition to Ms. Gillibrand, Senator Amy Klobuchar of Minnesota has drawn on her electoral success in red counties to position herself as a bridge-builder in increasingly polarized times. And Senator Elizabeth Warren of Massachusetts — who soundly defeated a popular Republican incumbent in her first election — has focused recently on addressing concerns that she’s simply an “ideas candidate,” combining her rhetoric about economic inequality with a more explicit pitch on her ability to beat Mr. Trump. (A fourth leading female candidate, Senator Kamala Harris of California, has enjoyed most of her success thus far in Democratic strongholds.)

As they now campaign for president, they are encountering some of the same misogyny that Mrs. Clinton faced when she ran in 2016. They are running up against assumptions voters and pundits have about what presidential leadership looks like, battling a presidential archetype where men are the only touchstones.

As a result, they are frequently asked to explain why they believe they have paths to victory, and prove they can win over prized working-class voters in critical states like Ohio, Michigan and Pennsylvania. This has come even as polls have consistently found that numerous Democrats — including multiple women — enjoy an early edge in head-to-head matchups against Mr. Trump.