By Alex Rogers, CNN

(CNN) -- The brawl over Justice Ruth Bader Ginsburg's successor began before her body was buried. Within hours of her death on Friday, Senate Majority Leader Mitch McConnell said, "President Trump's nominee will receive a vote on the floor of the United States Senate."

Since then, almost every Republican senator running for reelection has announced their support for the process to confirm the to-be-announced nominee, shaking up the race for Senate control only six weeks out from Election Day.

The quick embrace of McConnell's strategy could help Republicans in toss-up Senate races in purple and red states, including in North Carolina, Iowa, Georgia and Montana, although it could hurt them in others.

North Carolina Sen. Thom Tillis has struggled to coalesce the Republican Party behind him, with polls showing Trump leading him in the state. But the battle over the Supreme Court could endear him to the President's supporters.

"I think what it does is it helps him with the Trump voters who might not vote down ticket," Glen Bolger, Tillis' pollster, told CNN. "But now suddenly see, 'Oh, the Senate's an important fight.'"

But Tillis may also have to satisfy those who question why his position has changed since 2016. The first question he received in the Senate debate on Tuesday was whether he flip-flopped on how to handle a Supreme Court vacancy.

In 2016, Tillis, like almost every other Republican senator, supported blocking President Barack Obama's nominee, Judge Merrick Garland, from filling the seat left by the late conservative Justice Antonin Scalia. But at a Fayetteville rally with President Donald Trump on Saturday, Tillis announced to cheers from the crowd that he would vote for whoever Trump picked from his list to replace the late Ginsburg.

While both nominees were chosen in the final year of a presidential term, Tillis said on Tuesday that there was a difference: Obama was a "lame duck" in 2016, while Trump, who is running for reelection, "deserves" to put forward his nominee.

Tillis then attacked his Democratic opponent Cal Cunningham on Tuesday for supporting Democratic presidential nominee Joe Biden, who the senator said would nominate "radical left, activist judges that would be wrong for this country." Cunningham responded that Tillis wrote a "blank check to the president" in supporting a Trump pick before it was even announced, when he should have been a "check and balance."

Republicans argue that confirming a sixth conservative justice to the court could help them in other states too, pointing to Senate GOP victories in Indiana and Missouri after confirming Judge Brett Kavanaugh to the court in early October 2018. That year, then-Missouri Democratic Sen. Claire McCaskill acknowledged the vote was "not a 'political winner'" for her. She voted along with every other fellow Democrat against Kavanaugh, but lost by fewer than six points.

Ryan Koopmans, a former top aide to Iowa Gov. Kim Reynolds, said the Supreme Court vacancy could help Republican Sen. Joni Ernst in her race against Democrat Theresa Greenfield, noting that Iowa has a record of supporting conservative judges. In 2010, it voted to remove three justices from the Iowa Supreme Court who ruled in favor of same-sex marriage.

"I think that this issue has the potential to start to drown out others in the coming weeks," Koopmans said.

If Sen. Doug Jones loses in deep red Alabama, Democrats need to win four seats and the White House to take control of the chamber. Republicans are worried that the fight over Ginsburg's successor could hurt Republicans in at least two blue states, Sen. Susan Collins in Maine and Sen. Cory Gardner in Colorado.

Gardner has said he would vote to confirm a "qualified nominee." But Collins has said that she will oppose any pick due to its proximity to election. She said her position is "in the interest of being fair to the American people -- and consistent" with what she did back in 2016.

Collins' decision could help her regain the support of those who have appreciated her independent stances in the past, like opposing the Republican effort to repeal the Affordable Care Act in 2017. But it will not be enough for some of her former supporters, who say that she changed during the Trump era and abhor her support of Kavanaugh.

"I have no faith in Susan to do the right thing, like I've counted on for so many years," said Carroll Payne, a Democrat and former Collins supporter running for a state House seat.

Democratic Senate candidates across the country have pushed to fulfill Ginsburg's "most fervent wish" to not be replaced until after the next presidential inauguration, when a President Joe Biden could choose Ginsburg's replacement. They have no power to block a unified Senate Republican majority, but are warning voters that a Supreme Court with six conservative justices could dismantle the ACA and its protections for those with pre-existing conditions. The Court hears a crucial case on the law's constitutionality a week after Election Day.

Montana Gov. Steve Bullock launched an ad on Monday saying that his Republican opponent, Sen. Steve Daines, voted to strike down the ACA, while he would protect citizens from insurance companies denying people coverage for those with pre-existing conditions.

The National Republican Senatorial Committee aired an ad hitting back at Bullock, saying if he's elected, Democrats in charge of the House, Senate and White House could "change Montana overnight," leading to "liberal judges, stricter gun laws and government-run health care."

The Democratic party has become more energized on the Supreme Court issue during the Trump era. Many Democrats view the seat that Justice Neil Gorsuch now occupies, instead of Garland, as stolen. They say that Kavanaugh's confirmation amid accusations of sexual assault -- allegations Kavanaugh denied -- delegitimized the court. And after Ginsburg's death, Senate Democratic Leader Chuck Schumer said that "nothing is off the table for next year" if Republicans replace her and he becomes majority leader.

But the Democratic candidates who could give Schumer the gavel have tried to separate themselves from the party's left wing. The campaigns for Jones, Cunningham, Jon Ossoff in Georgia and Mark Kelly in Arizona have indicated that their candidates do not support expanding the Supreme Court.

"North Carolinians are already voting and will continue to do so in the coming weeks," said Cunningham in a statement. "They deserve that opportunity to have their voices heard, and then, it should be up to the next President and next Senate to fill the vacancy on our Court."