Former U.N. Ambassador Nikki Haley, facing an uphill battle heading into the Republican nominating contest in South Carolina, said the race is "just getting started" and appeared undeterred by polls that show former President Donald Trump leading in her home state. 

"I just got to South Carolina," Haley said in an interview on "CBS Mornings" on Tuesday. "We just left New Hampshire. My goal has always been to keep building."

For the former South Carolina governor, the Palmetto State marks a major opportunity, after a third-place finish in Iowa and a loss to Trump by more than 10 points in New Hampshire earlier this month. And with a quirk in next week's contest in Nevada, where Trump and Haley won't go head-to-head, the Feb. 24 South Carolina primary marks the next major showdown. 

Though voters have familiarity with Haley, who served as the state's governor from 2011 to 2017, the conservative electorate is favorable for Trump. But Haley is aiming to come in even stronger in South Carolina than in the Granite State. 

"They know that I was a good governor," she said. "Now I have to show them I'll be a good president."

Still, a slew of endorsements of her opponent by GOP state officials has added to the headwinds for Haley, throwing her prospects in the state into question. 

Even so, Haley touted a handful of fundraising booms following an outburst by Trump — what she called a "temper tantrum" by the former president after her second-place showing in New Hampshire — as evidence that voters want an alternative.

"People don't want a coronation," she said. "It's been two states."

Haley reiterated her pitch for a new generation of leadership, saying that Trump and President Biden aren't as mentally fit as they used to be. 

"Are we really in this country going to have two 80-year-olds running for president?" Haley said. "The party that goes and puts a new generational leader in is the party that will win. That's why I'm running."

The former U.N. Ambassador also pushed for strikes against Iran following the death of three U.S. troops over the weekend, saying "this is not about playing hard at Iran is about playing smart," while advocating for "surgical" action to go after leadership.

Kaia Hubbard

Kaia Hubbard is a politics reporter for CBS News Digital based in Washington, D.C.