January 7, 2016 6:16 AM EST

Hillary Clinton has a simple new strategy to win the White House. “At the end of the day, this really comes down to whether I can encourage and mobilize and turn out women to vote for the first woman President,” Clinton told me late last year, in an interview for my new book, Broad Influence: How Women Are Changing the Way America Works. “I’m going to do my best to make that case.”

It’s hard to imagine Clinton uttering those words in her last campaign. Her pollster and chief strategist back then, Mark Penn, was convinced that America wouldn’t elect a woman unless she campaigned with a tough-talking macho exterior and avoided gender-based appeals. Her campaign brass advised Clinton against giving a speech on gender, even after Barack Obama delivered lauded remarks on race. Much of what would have gone in that never delivered address–calling the White House “the highest, hardest glass ceiling”–was spoken only when she conceded defeat.

This time around, after four years as Secretary of State with polls giving her high points for leadership qualities, Clinton is betting she can take advantage of a country on the cusp of a sea change. Studies have found that when women make up somewhere between 20% to 30% of any traditionally male-dominated body, whether it be a legislature, a corporate board, a Navy ship or an appellate court, they begin to change the way things are done. Women now account for 20% of Congress, 30% of upper-level civil service and political appointees and 35% of the federal bench. The final frontier is the executive office suite. Though women regularly outnumber men at the polls, only six of the nation’s 50 governors are women, along with just 18% of mayors.

To break through, Clinton must get a broad cross section of women to vote for her. And while she does well with older women, she has long struggled among younger and unmarried women, who voted as a group for Obama in 2008, swinging 16 primaries in the process. This year, her approval among young people, defined as 18-to-34-year-olds, is upside down, with 44% saying they are favorable and 48% saying they are unfavorable.

To win over these voters, who tend to like a change message, Clinton has begun to argue that electing a woman will be a transformation in itself. “There are some areas where our own life experiences really prepare us to be more receptive,” Clinton told me on Oct. 8. “I just think women in general are better listeners, are more collegial, more open to new ideas and how to make things work in a way that looks for win-win outcomes.” On the trail, Clinton frequently cites the fact that women earn more than 50% of college degrees and 60% of graduate degrees in the U.S. Now they need to spread through all levels of the workforce. “The private sector is much more impervious to public pressure,” Clinton says. “Therefore we have to create … a program that can provide that push to get more private-sector involvement in hiring and promoting women to positions of responsibility.” By 2030, when the baby-boomer generation fully retires, the workforce will be short 26 million workers. There are only two ways to bridge this shortfall: new immigration and bringing women fully into the workforce.

One model that interests Clinton is a six-year-old government program in Australia that asks companies to publicly report the progress they are making in hiring and training women and keeping them after maternity leave. “I think [a public-private partnership] is a very good idea,” Clinton says. “I do want to encourage and find ways to incentivize corporations to promote more women and to have more women on their boards of directors.”

So here’s the test: at a December Clinton town hall in Mason City, Iowa, first-time voters Susan Straka, 19, and Ali Reil, 20, both said they planned to caucus for Bernie Sanders. But they were not yet certain. “It depends what’s happening in the room. If she’s losing, I’ll caucus for her,” Reil added. “She’s a woman, and that’s a big part of it.”

This appears in the January 18, 2016 issue of TIME.

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