Atheists, agnostics and others religiously unaffiliated say their time is about to come in American politics, insisting that candidates from the federal to the local level will soon have to start trying to win them over.

Buoyed by poll numbers showing religious “nones” growing as a percentage of the U.S. population, the Secular Coalition for America is even trying to highlight candidates who are “nontheists” and says Democrats and Republicans will have to work to win their votes.

“I personally think that this is going to be the last presidential election where we are going to not hear more about the nontheist community,” said Larry Decker, executive director of the group.

Mr. Decker’s group has been highlighting a “secular candidate of the week” — an effort he likened to the American Humanist Association’s formation of the Freethought Equality Fund PAC to start spotlighting and endorsing openly nontheistic candidates.

Last week’s candidate was Misty Plowright, a transgender woman running as a Democrat for Congress in Colorado. The group quotes her as saying that “freedom and equality for all belief, including secular ones, are of the utmost importance to me.”

Mr. Decker likened his effort to groups that promote pro-choice or openly gay candidates.

“I wanted to make sure that we were letting other nontheistic people out there know that there are open atheist and agnostic and humanist candidates who are running for office,” he said.

The share of religiously “unaffiliated” people in the country — atheists, agnostics or “nothing in particular” — increased from about 16 percent in 2007 to about 23 percent in 2014, according to a Pew Research study last year.

A Pew poll this year found that religious “nones” make up one-fifth of all registered voters in the country — about in line with the percentage of white evangelical Protestants, who comprise a crucial piece of the Republican coalition.

Dave Campbell, a political science professor at the University of Notre Dame, said the secular crowd already plays a “strong but quiet, or largely unacknowledged role” in politics, “particularly within the Democratic coalitions.”

Many supporters of Sen. Bernard Sanders during the Democratic presidential primary contest were “highly secular,” said Mr. Campbell, who has written extensively about the intersection of religion and politics.

“They didn’t necessarily put opposition to the mixture of church and state as their top issues. Their top issues would be more economic,” he said. “But if you look at their actual belief system, they’re much more likely to be secular than are Hillary [Clinton] supporters, and of course they’re a very important part of the activist base of the Democratic Party.”

In the Pew poll this year, more than a quarter of Democratic and Democrat-leaning voters were religiously unaffiliated. Overall, about two-thirds of the unaffiliated said they supported Mrs. Clinton over Republican presidential nominee Donald Trump — about in line with the percentage who supported President Obama at the same point in 2012 over Republican nominee Mitt Romney.

Mr. Decker said religiously unaffiliated voters are more unified in support of pro-choice and same-sex marriage stances than evangelicals are in their opposition to those policies.

David Lane, a social conservative activist who has sought to recruit pastors to run for political office, said the country has lost its Judeo-Christian heritage to a certain extent and that it’s up to religiously inspired voters to turn out and vote if they hope to reverse that trend.

“If we stay home, somebody’s values are going to reign supreme,” Mr. Lane said. “You’re putting in people, secularists, who have no biblically based foundation, and they’re going to impose their radical social agenda on Americans who work, who go home, who teach Sunday school, who coach little league. These are radicals.”

The Secular Coalition has issued voter guides for the presidential candidates, giving Mrs. Clinton an A and Mr. Trump, who has aggressively courted evangelical voters, an F on their issues. Green Party nominee Jill Stein got an A, and Libertarian Party nominee Gary Johnson got a B.

Mr. Decker said that in the past year or so, several members of Congress told him, “You know, Larry, I’m not really a believer.”

“But they can’t say that publicly,” he said.

In 2013, Kyrsten Sinema was sworn into the House with her a hand not on a Bible but on a copy of the U.S. Constitution. The Arizona Democrat’s campaign told The New York Times in 2012 that she advocated a secular approach to government but that the terms “nontheist,” “atheist” and “nonbeliever” weren’t befitting of her work or character.

Pete Stark, a longtime California Democrat in Congress who lost his re-election bid in 2012, came out as an open nonbeliever in 2007 in response to an effort from the Secular Coalition to identify the highest-level “atheist, agnostic, humanist or any other kind of nontheist” holding elected public office in the U.S.

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