(CNN) -- A growing number of GOP Senate contenders are spurning former President Donald Trump's wishes with campaigns that defy his criticism or ignore his support of a rival, as the Republican Party debates his role in its future.

In Alabama, Trump's early endorsement of Rep. Mo Brooks has failed to stop the GOP primary field from expanding. Last week, Katie Boyd Britt, the former CEO of the state's Business Council, entered the race for the open US Senate seat, joining a field that includes Brooks and Lynda Blanchard, who served as US ambassador to Slovenia during the Trump administration.

In North Carolina, former Gov. Pat McCrory and former Rep. Mark Walker have declined to pull the plug on their campaigns -- treating Trump's surprise endorsement of Rep. Ted Budd at the state's GOP convention earlier this month as a minor obstacle as opposed to a campaign-ending blow.

And in Alaska, Sen. Lisa Murkowski is running for reelection even though Trump has vowed to campaign against her, calling her "disloyal" after she voted to convict the former President for inciting the insurrection at the Capitol in January.

The increasing number of Republican Senate hopefuls who have deferred their political futures to Trump has raised new questions about the former President's influence, right as he weighs a comeback bid in 2024 that aides say will hinge, in part, on how well Trump-backed candidates perform in the midterm elections next fall.

Rather than dropping out or reconsidering the viability of their campaigns, GOP candidates who have endured Trump's wrath or missed out on his endorsement are taking a different approach. While aligning themselves with Trump on various issues, they are also refraining from supporting his false claims about the 2020 election -- hoping the two-track approach will afford them a primary win and position them as the strongest candidate against their future Democratic opponent.

"Trump's presence looms over every race in ways that have a profound effect on the field, eliciting allies and rooting out foes, but the nod is diminished when every Republican is effectively running as a Trump-aligned candidate," GOP strategist Liam Donovan said.

North Carolina candidates press on despite losing Trump's endorsement

Trump privately gave Budd his endorsement about 15 minutes before he took the stage in Greenville, North Carolina, earlier this month, immediately upending the primary for the crucial 2022 Senate race. Budd's two main competitors -- McCrory and Walker -- have stayed in the race, undaunted that the most powerful figure in the Republican Party chose another standard bearer. In their response to the former President's announcement, both men chalked it up as a premature decision based on poor guidance Trump received from his team.

Walker said "it was tough" watching his girls cry during Trump's speech, while McCrory accused former White House chief of staff Mark Meadows of giving Trump "bad advice." In a tweet following Trump's endorsement, McCory added that he was "disappointed" by the development, saying Budd had "done more to oppose the Trump agenda than anyone in this race," pointing out the congressman's votes against a border security bill and hundreds of billions of dollars in pandemic relief that the former President supported.

"Pat McCrory is clearly the most electable candidate Republicans can put forth," said McCrory political consultant Paul Shumaker. "Republicans will not be able to win a majority in the United State Senate without winning North Carolina."

Republican Sen. Thom Tillis told CNN that while "endorsements can give you a bump," other factors, such as "your priorities" and "the issues" of the campaign, will prove decisive.

But Jonathan Felts, Budd's political adviser, told CNN that the other candidates' response to Trump's endorsement could cost them with GOP primary voters.

"Both of them have suggested that President Trump doesn't really know who supported his America First agenda," Felts said. "It appears their new strategy in a GOP primary is to insult President Trump. Good luck with that."

Trump's inability to sway Republicans 'could become a serious issue' ahead of 2024

In Alabama, Brooks became the front-runner for Senate after earning Trump's support for leading the charge in early January against Congress' certification of the 2020 election. That was before Britt, a former top staffer to the retiring Sen. Richard Shelby, entered the race last Friday -- quickly earning her old boss' approval and setting up a proxy fight between Trump and the senior senator's allies in the business community.

In the deeply red state, where the primary will virtually ensure Alabama's next senator, Britt has only had positive things to say about the former president, despite lacking his endorsement.

"President Trump accomplished so many tremendous things for our nation," Britt told CNN in a statement. "In the Senate, I will continue to advance the America First agenda by fighting to secure our Southern border, stand up to China, stand with Israel, protect the unborn and support conservative, commonsense economic policies that grow good-paying jobs and opportunity for all Alabamians -- no matter where in our great state they live."

A person involved in Trump's post-presidential operation said his inability to effectively clear the field in both Alabama and North Carolina was "a potential red flag," particularly as he weighs endorsements in other high-profile Senate primaries, such as those unfolding in Ohio and Missouri.

"He doesn't seem to have the other candidates running scared and that could become a serious issue later on," this person said.

Trump and his team had already been eying a summer rally in Alabama -- where he would appear alongside Brooks -- as one way to ignite momentum behind the outspoken conservative congressman before Britt entered the field. And while the process of selecting a location and planning the event had stalled in recent weeks, two people familiar with the effort said Shelby's glowing endorsement of Brooks' opponent revived the planning. "She's like family," Shelby told Politico. "She's probably the best-qualified candidate to come along in a long time. I'd support her, I'd vote for her."

"It definitely lit a fire under their asses," one of the people familiar with Trump's rally plans said of his team. A spokesperson for Trump did not immediately respond to a request for comment.

In a statement to CNN, senior Trump adviser Jason Miller said the former President's support remains coveted in GOP primaries and has already boosted the profiles of candidates who have received a Trump endorsement.

"Everybody wants President Trump's endorsement because it's an automatic catapult into first place, from Max Miller in Ohio to Mo Brooks in Alabama to Ted Budd in North Carolina," Jason Miller said.

Trump has often bragged about the potency of his endorsements, once claiming to have an "88 and 2" win-to-loss success rate in 2020 GOP primaries. But while the former President has mostly chosen successful candidates in the past, the current slate of races where his maneuverings have failed to trigger dropouts comes against a different backdrop. Mulling a potential White House bid in 2024, Trump and his aides have come to view 2022 as fertile ground to test his appeal and influence with the GOP base. People close to the former president told CNN it is highly unlikely that Trump will make a firm decision about 2024 until he's seen the outcome of next November's congressional elections. Trump has been using the 2022 midterm cycle to flex his muscles as a GOP kingmaker and assist Republicans in their quest to regain control of the House and Senate.

One of the people close to Trump said the former President "knows (that) 2022 will put his influence to the test" and either confirm his grip on the Republican Party or bolster calls for him to step aside for fresh talent.

Trump's sway will also be under scrutiny in Alaska, where Murkowski is waging a reelection bid despite his pledge to oppose her. Murkowski supported impeaching Trump during the Senate trial following the January 6 riot at the US Capitol, saying he "failed" to uphold his oath to "defend America and all that we hold sacred." Kelly Tshibaka, a former Alaska Department of Administration commissioner, is running against Murkowski with the aid of several members of Trump's 2020 campaign team, though Trump has not yet offered Tshibaka his endorsement.

To be sure, Trump still holds power over Republicans seeking higher office. Last week, Arizona, state attorney general Brnovich entered the Senate race against Jim Lamon, a solar energy entrepreneur, and Retired Maj. Gen. Michael "Mick" McGuire even though Trump had blasted him for not supporting his false claims of widespread election fraud in the state.Lamon has taken to the airwaves in a bid for Trump's attention and possible endorsement, airing a TV ad on enhanced border security in New Jersey, where Trump is spending the summer at his Bedminster golf club.

Lamon spokesman Stephen Puetz told CNN, "Mark Brnovich has never had President Trump's back -- which is why President Trump has made his feelings about Brnovich so clear."

"Arizona needs a fighter in Washington who will stand up for conservative values and that's Jim Lamon," Puetz added.

But Brnovich has said he's spoken with Trump since the former president criticized him and on Monday the Arizona attorney general told the Justice Department his office would "not tolerate any effort to undermine" the state Senate's controversial audit of its most populous county. His campaign took on Biden's presidency in a statement.

"Attorney General Brnovich is no stranger to attacks from the left, but it is sad that we are seeing politically motivated disinformation coming from unvetted members of his own party just days after he launched his campaign for US Senate," Brnovich campaign spokesman George Khalaf said. "Attorney General Brnovich will continue to fight for Arizonans, uphold the rule of law, and push back against the Biden administration's gross executive overreach."

CNN's Manu Raju and Michael Warren contributed to this report.

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