Sara Gordon cooed at dogs, rapped on doors and dropped off fliers declaring Mitt Romney the “king of greed” for more than an hour before she finally found who she was looking for: Laura Barina, not a union member, who voted for Barack Obama in 2008 but isn’t so sure she will again.

“Obama makes me nervous. He’s been in for four years. Why hasn’t he done more?” Barina, 58, asked Gordon as she used a slippered foot to tap down muddy divots in her front lawn.

“And do you feel Mitt Romney is going to be more effective?” Gordon responded gently. “Well, I don’t know,” Barina said. “He scares me, too. He doesn’t understand people like us, how we live.”

Gordon works for the American Federation of Teachers in Buffalo, and she’s in Ohio as part of an effort to bolster Obama, partly by taking advantage of new rules that allow union members to more aggressively go after non-union voters.

Best known for opening a new spigot of anonymous corporate giving in elections, the Supreme Court’s Citizens United v. FEC decision in 2010 also allowed labor unions to use their general treasury cash, including dues, to reach out to all voters, not just union members. In Ohio, the list of potential voters whom the AFL-CIO will reach out to in the last month before the election has nearly doubled, from 1.2 million in 2008 to 2 million this year.

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Union officials hope their new freedom will prove especially potent in Ohio, where a year ago voters repealed a new law pushed by Republican Gov. John Kasich that would have restricted collective bargaining rights for public employees, including police officers, firefighters and teachers.

The 62 to 38 percent referendum vote — a surprisingly lopsided figure in a state with a history of razor-thin election margins — came after the AFL-CIO waged an all-out battle against Kasich and the law, in the process building a massive list of non-members who seemed friendly to their positions.

“We had nine months last year to educate,” said Tim Burga, president of the Ohio AFL-CIO. “It really opened some eyes in the state of Ohio — not just inside of labor but with the general public — about what’s happened to the Republican Party, that it’s been taken over by extremists.”

The unions have also worked to capitalize on state support for the federal bailout of GM and Chrysler, a popular issue in Ohio, which is home to key auto parts manufacturers.

Those issues, combined with the Citizens United ruling, help explain why Obama has held a steady lead in Ohio, union leaders say, even as the race has fluctuated some on the national level.

A new poll released Tuesday by CNN/ORC showed Obama maintaining a four-percentage point lead, 51 to 47, in Ohio at the same time Romney has moved ahead in a handful of national surveys.

Ohio Republicans believe there will be little carryover from the 2011 collective-bargaining fight to this year’s election. Last year’s vote was a state issue, they argue.

More important for predicting this year’s outcome was a federal initiative on the same ballot, when Ohioans voted 66 percent to 34 percent to send a message of opposition to Obama’s health-care law by declaring that the law’s insurance-purchase mandate shouldn’t be legal in the state.

“It was a federal issue,” said Scott Jennings, Romney’s Ohio state director for the health-care vote. “It’s germane to this debate. The other one, not so much.”

But union workers are doing their best to tie last year’s fight to the national conversation. They remind voters that last October Romney spent 45 minutes at a Cincinnati phone bank for Republican volunteers urging voters to retain the law.

“I fully support Gov. Kasich’s Question 2 in Ohio,” Romney said a day later, declaring himself “110 percent” behind the law after coming under fire from his GOP primary opponents for appearing to refuse to take a position while standing beside the volunteers.

Romney has endorsed a national “right to work” law, which would make it illegal to require workers to join unions across the country, an issue popular with the Republican base but anathema to unions.

And just last week, he included references to “card check” legislation into stump speeches in Denver and Fishersville, Va., the first time in months that he’d mentioned his opposition to the pro-union legislation largely abandoned by Democrats.

“I’ve got guys right now in the firehouse in recliners watching Fox News,” said Mike Norman, a longtime Cleveland firefighter who is secretary of the city’s firefighters union. “Not all, but some, are now realizing — perhaps begrudgingly — that Barack Obama is the candidate who’s looking out for them and who is closer to their reality.”

The Ohio law last year targeted firefighters and police officers, along with teachers and other public employee unions. Republicans now argue that was a tactical mistake, allowing the unions to rally public support.

The record was far more mixed for unions in Wisconsin, where a massive effort to recall Republican Gov. Scott Walker for cracking down on collective bargaining with a law that excluded police and firefighter unions fell flat.

“I think they’re trying to catch lightning in a bottle again,” said Greg R. Lawson, a policy analyst at a conservative Columbus-based think tank, the Buckeye Institute for Public Policy Solutions. “They’ll have an influence; they always do. But they want to make it out like there’s this huge revolution that’s occurred, and I don’t get a sense that’s happened. ”

In Fairview Park, Barina told Gordon that she supported repealing the Republican law last year. “I know a lot of people in the unions,” she said. “And my son’s a teacher.”

Still, she said, she’s torn about her presidential vote.

Walking down the sidewalk to her next target, Gordon said she felt good about the encounter. Barina might not be ready to vote for Obama yet, but she said she supports continuing Medicare without changes. She seemed open and she might be persuaded by a phone call in the near future.

“I think she’s less undecided than she thinks,” Gordon said.