Primary results showing Edwards in a fight for third place was a vast improvement over his single-digit standing in the polls immediately after his second-place Iowa finish, he said.
"It's important for me to show I can move up," Edwards said.
Edwards had hoped for a top-three finish in Tuesday's primary. He was in a tight race with Gen. Wesley Clark for a distant third Tuesday night, well behind Massachusetts Sen. John Kerry's first place and former Vermont Gov. Howard Dean's second.
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"New Hampshire has been very good to a visitor from North Carolina," he said at a rally. "Look at what's happened over the last few weeks. We went from five to 32 percent and a strong second-place finish in Iowa, and we are very proud of that."
The North Carolina senator said he hopes to win South Carolina -- his birth state -- adding that he has the votes down there to do it.
"In New Hampshire 10 days ago, we were 20 points behind General Clark, and look at what we've done," he said. "And now we're going to take this energy and momentum we saw in Iowa and this energy and momentum we saw in New Hampshire, and we're going to take it right through Feb. 3rd."
Despite spending a good part of the past year campaigning in New Hampshire and holding more than 100 town hall meetings, Edwards could not overcome the built-in advantages of the New Englanders.
"They're from right next door," Edwards said of Kerry and Dean. "They're expected to do that."
Edwards wants to make a stand with a win next week in South Carolina, his native state. Though he said he must win there, he brushed off questions about whether a loss would end his candidacy.
"Analysts were saying three weeks ago, I shouldn't be in the race in Iowa," Edwards told CNN. "Here 10 days ago, I was in low, single digits, and you look at what's happened.
"What this means is this campaign and this message is working," he said. "And so I feel very good about South Carolina and these subsequent states."
Exit polls showed Edwards was most competitive among those who thought health care, education and the economy were the top issues -- getting almost one in five among those voter groups.
Edwards declared himself the positive, optimistic candidate. But he split the vote among voters who thought having a positive message was the most important candidate quality.
He was relatively strong among those who most wanted a candidate who cares about people like them, but slightly behind Kerry and Dean with that group.
The exit polls were conducted for The Associated Press by Edison Media Research and Mitofsky International.
His tight race with Clark could portend another close contest next week when the campaign turns to seven races on Feb. 3, including South Carolina, considered all important to Edwards' chances.
He was born in South Carolina and represents next-door North Carolina. But Clark also wants to claim the mantle as the candidate of the South.
The Edwards campaign also wants to do well in Oklahoma and possibly New Mexico and Missouri. TV ads are airing in South Carolina, Oklahoma and New Mexico. Edwards said contributions continued to flow into his campaign based on his Iowa finish.
Edwards said he would work hard in the coming week but would not forecast his prospects.
"Beyond South Carolina, I don't want to make any predictions, but I want to do well," he said.
Edwards was scheduled to fly to South Carolina after speaking to supporters in New Hampshire. He planned to spend part of Wednesday campaigning in South Carolina, as well as in Oklahoma and Missouri -- a big prize next week that became competitive after favorite son Rep. Dick Gephardt dropped out after the Iowa caucuses.
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