The clamor for these women has spread beyond Silicon Valley’s large public tech companies to privately held start-ups as well, some of which could emerge as the next public giants. Glenn Kelman, the chief executive of Redfin, a closely held online real estate brokerage, said he sometimes had to mount “pitched efforts” to attract female candidates to his company’s nine-member board, because many other start-up chief executives were after the same people.

“A lot of executives are trying to do this,” Mr. Kelman said. “If I dillydallied, suddenly everyone was recruiting” the same person.

In November, Redfin brought on two new directors, including Julie Bornstein, the chief operating officer of Stitch Fix, an online personal stylist service. Ms. Bornstein is the second woman serving on Redfin’s board. In 2014, the company added Ms. Tobaccowala, a co-founder of Evite who was later a top executive at the online polling company SurveyMonkey.

Ms. Bornstein, 46, a former chief digital officer at Sephora, said she had gotten 30 or 40 calls to discuss joining a corporate board over the last few years, even though she had never sat on one. Before becoming a member of Redfin’s board, Ms. Bornstein said, she also talked to two public companies about a board spot, though she lost out to other women for those positions.

Yet many hurdles still prevent women from joining more tech boards, Ms. Bornstein said. As she interviewed at companies about board seats, she said, she found that many directors typically stuck to their own networks of acquaintances for candidates. That generally led to looking at the same people over and over again.

As a rule of thumb, boards also want experienced members. Part of the reason Ms. Bornstein said she did not land other board positions was because “they couldn’t get their head around someone who hadn’t been on a board before.” Because there are “fewer natural high-profile women in tech,” she said, “it requires taking bets on new people.”

Efforts are afoot to break down those risks and bring new candidates to attention. Sukhinder Singh Cassidy, the founder and chairman of Joyus, a video and e-commerce start-up in San Francisco, started an initiative called theBoardlist in mid-2015 to highlight women qualified to be on boards. To be on the list, a candidate must be endorsed by a Boardlist member, many of whom are venture capitalists, tech executives and others.