Retired neurosurgeon Ben Carson has surged to the No. 2 spot in the GOP field, overtaking former Florida Gov. Jeb Bush and, combined with front-running businessman Donald Trump, gives outsiders a firm grip atop the Republican presidential race.

Combined, the two men have 40 percent of the vote in the latest Quinnipiac University Poll. Add in businesswoman Carly Fiorina’s 5 percent, and nearly half of Republicans say their No. 1 choice for president is someone who has never won a public election.

But it’s Mr. Carson’s swelling support that stands out.

“He’s the safe rebel,” said Michael McKenna, a GOP strategist and pollster.

Despite a debate performance that garnered solid but not overwhelming praise, and an August devoid of high-profile television feuds that much of the rest of the GOP is engaged in, Mr. Carson overtook Mr. Bush in the RealClearPolitics.com average of polls as of Thursday. Quinnipiac’s poll was a big part of that boost, giving Mr. Carson 12 percent support, which is significantly more than Mr. Bush, who’s fallen to 7 percent, tied with Sens. Ted Cruz and Marco Rubio.

John Philip Sousa IV, who is running The 2016 Committee, a super political action committee trying to elect Mr. Carson, said grass-roots work and word-of-mouth are helping the retired doctor, and he credited thousands of volunteers he said are spreading the word without the sort of fanfare that’s accompanied Mr. Trump’s media-saturated rise.

“They’re quiet, they’re efficient, and they get the job done,” he said.

The anger of the GOP primary electorate has been palpable for years, and led to tea party-backed candidates ousting incumbents or party favorites in a series of 2010 and 2012 primaries. Voters also seemed to repel from Mitt Romney, the establishment-backed pick for the GOP presidential nomination in 2012 — though he outlasted the field and claimed the title, losing eventually to President Obama in the general election.

Analysts said they’re waiting to see whether this year’s crop of anti-establishment types can hang on in a Republican field freighted with heavyweights who bring decades of experience.

“It’s still the first or second inning, and pitchers look a little less invincible the third time through a lineup,” said Kevin Sheridan, a strategist who worked for the Romney campaign. “Let’s see if any of them can maintain those levels over the coming weeks.”

He said candidates like Mr. Carson will also have to show they can raise money and build political operations that can translate poll support into primary or caucus votes, and there’s not been much sign of that yet. Still, he said there has been a real change, with a significant number of voters rejecting candidates they see as “more of the same.”

“It used to be how well a candidate answered the question of ‘How can you win?’ In 2015 it’s ‘How are you any different?’” Mr. Sheridan said.

For now, though, all three of the outsiders are showing well in early primary states.

David Starr, chairman of the Northern Grafton County Republican Committee in New Hampshire, said Ms. Fiorina drew a crowd of about 150 to a recent event, and “she came out of there probably with 150 primary votes.”

Mr. Starr said Ms. Fiorina exudes substance, while Mr. Trump is a showman, and Mr. Carson appears to be a nice guy, though how his soft-spoken doctor’s bedside manner will play is still to be seen.

“As to what’s going to happen, I don’t have a clue,” he said. “I know what I’d like to happen. I’d like to have the Republicans nominate somebody a little more electable than Donald. Carson would be good, any of the governors would be pretty good. That’s what I’d like to have happen. I’m not sure if it’s [going to] happen. The party has sort of a built-in death wish, going right back to, say, ‘64 when they nominated [Barry] Goldwater. And they’ve done similar dumbass things ever since.”

Mr. Carson spent a large part of August in Iowa, which is home to the first-in-the-nation caucuses, and was in New Hampshire and South Carolina, the next states to hold primaries, as well as stopping in Nevada and New York.

He paid a visit to the Colorado site of the Gold King Mine breach, where he said the Environmental Protection Agency’s failure to manage the project was emblematic of the agency’s broader failures. And he toured the U.S.-Mexico border near Douglas, Arizona, calling the failure to better secure the border a problem of willpower, not ability.

“I think Dr. Carson, the next step for him in solidifying those numbers, because he has some pretty impressive numbers right now, is to fill in the details on policy. And again, that’s different than saying in general-speak, ‘I support an individual policy.’ I mean, it is really filling in, ‘Here is my plan on the budget. Here is my plan on taxes. Here is my plan on energy,’” said Tim Phillips, president of Americans for Prosperity, a leading conservative advocacy group.

He said that goes for Mr. Trump and for Ms. Fiorina, who make up the triumvirate of candidates who’ve never held office and don’t have a voting record to point to.

Among the good news for Mr. Carson is that he has the highest net-favorable rating of any candidate Quinnipiac surveyed, and was last on the list of candidates Republican voters said they would “definitely not” vote for, with just 5 percent ruling him out.

Mr. Trump topped the “Not” list, with 26 percent saying they couldn’t ever support him — almost the same as the 28 percent who are backing him.

Pollsters have described the Trump effect on the race as unprecedented, and say it’s tough to guess what will happen — though it’s clear he has some staying power.

Mr. Sousa said Mr. Trump has exposed a large portion of the electorate fed up with Washington, but he said those fed-up voters will increasingly gravitate toward Mr. Carson, who Mr. Sousa said will offer less “extreme” policies.

“Like millions and millions of people, part of me cheers a little when Trump says some of those things, because we really need to take care of America first, and we need to put America back to work, especially in minority communities,” Mr. Sousa said. “But these extreme ideas probably aren’t going to get the job done, probably aren’t going to stand up in court, and will take forever to get implement[ed] — unlike the Carson program, that most people can get behind.”

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