Hillary Clinton delivers a speech in Toledo, Ohio. REUTERS/Brian Snyder Despite a clear victory for Hillary Clinton in the first presidential debate, a new set of polls shows that she's losing to Donald Trump in the key swing state of Ohio.

The Republican presidential nominee is ahead by several points there, according to a Quinnipiac University poll released Monday. In a four-way race among Trump, Clinton, Libertarian candidate Gary Johnson, and Green Party candidate Jill Stein, Trump has 47% of the vote, Clinton takes 42%, Johnson has 6%, and Stein takes 1%.

Clinton is, however, ahead in the other swing states included in the poll. She leads Trump by 5 points in Florida, 3 points in North Carolina, and 4 points in Pennsylvania.

The poll was conducted between September 27 and October 2. Each state poll has a margin of error of between 4.2 and 4.4 points.

"Although Hillary Clinton clearly won the first debate with Donald Trump, this victory did her only little good in her race for the White House," Peter Brown, assistant director of the Quinnipiac University poll, said in a press release. "Likely voters in Florida, North Carolina, Ohio, and Pennsylvania, the four largest and most important swing states, seem little closer to an Election Day decision."

Although several polls showed Clinton winning the first presidential debate by a wide margin, she hasn't seen a corresponding large bump in voter support.

"Many voters don't have to think a presidential candidate is a good debater to support their candidate in 2016," Brown said in the release. "In Florida, likely voters give Hillary Clinton a 35 percentage point margin for winning the debate, but only a 5-point margin in the election matchup."

Still, the Florida poll is good news for Clinton, considering that state was deadlocked early last month. Clinton also recently snagged the endorsement of NBA star LeBron James, who is arguably the most famous man in Ohio.

And other polls released recently show Clinton leading Trump in swing states.

As statistician Nate Silver pointed out on Twitter, state polls have been pretty consistent in predicting a Clinton win nationally.