New Congress live updates: What's next after Kevin McCarthy's speakership win

There is now an offer on the table from McCarthy to the small but influential group of hard-line conservatives opposing his bid for speaker, sources familiar with discussions told ABC News.

The Republican leader, who has now notched six successive losses for speaker, has proposed a new round of key concessions that includes making it easier for members to remove him as speaker.

McCarthy's offer, if approved by the full GOP conference on the floor in the rules package, would make it so a single member from either party could trigger an up-or-down simple majority vote on whether to oust the speaker. This is down from the threshold of five members McCarthy initially agreed to.

It’s unclear if the offer will make a difference, as some hard-line conservatives are digging in against McCarthy. The House is set to reconvene at noon today.

He’s also offered to put more members of the conservative House Freedom Caucus on the House Rules Committee and votes on legislation his opponents have been pushing for, including term limits for members of Congress and border security.

The group of conservative rebels who’ve opposed him are meeting this morning to consider the offer and plot their strategy going forward.

-ABC News' Rachel Scott, Katherine Faulders and Trish Turner

Leaving the chamber after it adjourned for the night on Wednesday, McCarthy -- surrounded by a phalanx of security and a mob of reporters -- initially joked and razzed a reporter he’s known for more than a decade. He seemingly celebrated a razor-thin GOP vote to adjourn the chamber after another day of unsuccessful attempts to elect him speaker.

He also said “continual votes," like they been having since Tuesday, are unproductive, instead touting the evening’s closed-door talks as the way forward.

McCarthy asked the reporter, John Bresnahan, “Club for Growth -- is that movement for you? You just saw a vote on the floor that the conference voted together on?”

McCarthy was referring to a newly announced deal between a political action committee aligned with him and Club for Growth, another major conservative group.

Bresnahan jokingly replied of the successful adjournment vote, “You’re very proud of that.”

“Hey, I crawl before I walk and walk before I run, and I felt as though we had a very good discussion," McCarthy said, adding, “I think what you should gauge, being able to do that vote, [is] that the discussions are going well but that doesn’t mean they’re done. That doesn’t mean anything on that. But it’s just in that realm.”

Another reporter asked, “What happens between now and noon tomorrow [when the House returns]?”

“We talk until we get this done,” McCarthy said.

Bresnahan asked if there would be votes tomorrow, to which McCarthy replied: “People know where everybody’s at. These votes don’t really change at all. So I mean, that’s fine. I think these discussions help that. I think from this aspect of seeing the vote here -- people want to spend their time discussing than being on the floor.”

A reporter asked if McCarthy has asked Byron Donalds, a GOP protest candidate, to drop out. McCarthy said, “No. No.”

-ABC News' Trish Turner

The motion to adjourn that was introduced shortly after the House returned from its Wednesday afternoon break passed by a narrow 216-214 vote. The chamber will return at noon on Thursday, presumably for more speaker votes.

Republicans had pushed for the adjournment, while Democrats and four Republicans opposed it, with some Democrats shouting for the clerk to stop the vote while the "no" votes were ahead.

The move to end for the night came after McCarthy held a closed-door meeting with detractors trying to block him from winning the speaker's gavel. The meeting yielded no deal, but McCarthy said progress was made.

The vote to adjourn marked the climax of a second day of historic failed speaker votes -- the first time in a century it has taken more than one round to pick a speaker.

Lawmakers gathered Wednesday around 8 p.m. in the House after their afternoon adjournment and swiftly launched into a vote to adjourn again until Thursday at noon.

The House clerk asked for a voice vote, with Democrats then pressing for an individual counting of the votes, which is underway.

The effort to adjourn comes after McCarthy said progress had been made with the group of GOP critics blocking him the speakership but that there was no path yet to a firm deal.