ABC News' Shushannah Walshe and Barbara Schmitt report:

Six-term GOP incumbent Sen. Thad Cochran dodged political death Tuesday by reaching out to Mississippi Democrats, including African-American voters, in the weeks between the primary and Tuesday's runoff. Turnout was even up, incredibly rare for a runoff that he now appears to have won.

But there were no congratulations from the apparent loser, no concession speech; instead, tea party-backed challenger State Sen. Chris McDaniel told his crowd of about 200 supporters Tuesday night in Hattiesburg: "The fight is not over."

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The former talk radio host followed that up today with a statement confirming that he will look into a possible legal fight, calling for "scrutiny of the election's irregularities," as well as a "thorough examination of the core principles of the Republican Party."

"In the coming days, our team will look into the irregularities to determine whether a challenge is warranted," McDaniel, 41, said of the allegations. "After we've examined the data, we will make a decision about whether and how to proceed."

In his speech Tuesday night, McDaniel said, "Before this race ends, we have to be absolutely certain that the Republican primary was won by Republican voters."

With 100 percent of precincts reporting, Cochran, 76, bested McDaniel 50.9 percent to 49.1 percent, a difference of fewer than 6,700 votes.

The Cochran campaign openly courted African-American voters, who are traditionally Democrats, but it is perfectly legal in the state. The primary is an open one, so any Democrat who did not vote in the Democratic primary earlier this month was free to vote Tuesday. Legally, a voter casting a ballot Tuesday must have had the intent to vote for the same candidate in the general election, something impossible to enforce.

The state GOP says counties are certifying votes through next Tuesday when the statewide total will be certified. The state GOP says there is a list of who voted in each primary, so if the McDaniel campaign wanted to confirm that no one who voted Tuesday also voted in the Democratic primary they could. 84,339 Democrats cast ballots in the Democratic primary June 3, which was not hotly contested like the GOP primary.

More than 360,000 votes were cast Tuesday. The McDaniel campaign hasn't approached the party about a possible legal fight, a party official told ABC News

Outside national tea party groups fighting for McDaniel spent about $7 million and several have expressed their anger at the results. FreedomWorks is one of the groups that dumped not only money, but activists flooding the state to knock on doors and get out the vote. They even sent poll observers to the state.

Adam Brandon, the executive vice president of FreedomWorks, told ABC News those poll observers are going through their notes to look for any irregularities.

Brandon said they had made no decision, but if there is a valid "legal opportunity" to get involved, they will.

"If you voted in the Democratic primary and you showed up in the Republican primary, that's a no-no," Brandon said. "If that happened there's a legal challenge right now. It's still early. If there is an opportunity to look at or challenge we would be committing malpractice if we didn't, but we only want to do it if it legitimate. So if it is legitimate, hell yeah. So we are checking everything out."

Not all groups have expressed eagerness to get involved legally. The Senate Conservative Fund, another group heavily backing McDaniel, told ABC News they would wage no legal battle.

Mississippi election law expert Matthew Steffey said a legal challenge is "possible in the sense anybody can sue anybody and all it takes is a filing fee and a lawyer, but I think saying the chances are slim overstates the chances of him successfully overturning the election.

Steffey, a professor of election law at Mississippi College School of Law in Jackson, detailed the barriers McDaniel would have to go through in order to claim victory, noting he would have to identify a "systemic problem so pervasive as to have the entire election thrown out," something he believes is without precedent.

While some states have mandatory recounts when the margins are narrow, that does not exist in Mississippi.

"There is a very specific procedure when ballots are challenged and the grounds for it are very limited," Steffey said, noting the challenge must be done in person as the ballots were being cast Tuesday.

There were poll observers on the ground, but Steffey has seen no reports of widespread challenges. He noted there is a law that no one can vote in a primary without the intent to support that candidate in the general election, but says it is "unenforceable."

"If you wanted to enforce it, it would have to be done when the ballots were cast," Steffey said. "A lot of the people who voted were registered Democrats, but in an open primary they are allowed to vote. They would end up having to target visible Democrats, which likely would have been a large number of African-American voters."

Steffey noted that given Mississippi's history with civil rights, that "would not be tolerated, not for a minute, but it certainly resonated with an unflattering piece of the past."

And as for Democrats' voting twice, Steffey points out if that were true, it would likely be caught at the polls. He noted McDaniel could run as a write-in, but that would be his only other pathway as it's too late to file as an Independent candidate.

"It seems like the early stages of grief, where denial and bargaining is still going on," he said.