Hours after the end of the combative presidential debate Tuesday, Mitt Romney’s campaign began running a striking new commercial that seeks to soften his positions on abortion and contraception through the voice of a supporter who said she voted for Barack Obama in 2008.

Before dawn Wednesday, Democrats had taken to Facebook, Twitter, Tumblr and television to ridicule Romney’s debate-night statement that he had collected “binders full of women” when he was a new governor in Massachusetts seeking “qualified” female appointees for his administration.

And on the campaign trail and on the air, the candidates and their allies argued intensely all day over who would do more to help women. At the same time, the topic of whether the combative encounter Tuesday night had alienated the very female voters they were seeking to connect with became fodder for cable TV discussions.

The level of intensity left little doubt that the election was coming down not only to a state-by-state fight for territory but also to one for the allegiance of vital demographic groups, chief among them undecided women.

For Romney, the imperative, with less than three weeks until Election Day, is cutting into what has been Obama’s sizable lead among women.

Their goal, Romney aides said, is to keep Obama’s lead among women, which in many polls has been in double digits, down to the low single digits.

Key to that effort, they said, is changing the perception among undecided women that Romney holds very conservative positions on social issues, after a Republican primary campaign in which Romney was under constant pressure from the right.

Social versus fiscal issues

Through polling and focus groups, the Romney campaign has found that while undecided women said they were concerned primarily about economic issues, they were troubled by whether Romney’s positions on issues like abortion and contraception were too extreme.

Romney, who while running for governor in 2002 said he would govern as pro-choice but subsequently came to shift his position, now opposes abortion rights except in cases of rape and incest. On access to contraception, Romney has emphasized his opposition to Obama administration policies that he says pressure religious employers to provide health insurance that covers contraception.

He has not sought to limit access to contraception but has voiced support for a provision that would give other employers the right to deny coverage for contraception on moral grounds.

Romney and his team have tried to address these concerns. They said perceptions of Romney’s positions had been unfairly shaped by Obama’s heavy advertising campaign, including ads focused on abortion rights.

A look at the ads

According to data from Kantar Media/CMAG, the Obama campaign and Democratic groups have run commercials relating to abortion about 30,000 times since July 2 – about 10 percent of their ads – including one that falsely claimed Romney’s opposition to abortion extended to cases of rape and incest.

Romney’s latest television ad answering that barrage potentially creates the risk that it would remind voters of how Romney has altered his position on abortion over the course of his political career.

Romney campaign strategists said they decided to release the ad this week because they believe it will have maximum impact as late-deciding voters tune in.

The ad was produced within the past two weeks, after strategists identified Sara Minto of Ohio, one of their volunteers, as a plausible “ideal everywoman.”

“You know, those ads that say Mitt Romney would ban all abortions and contraception seemed a bit extreme, so I looked into it,” Minto says, sitting in her living room. “Turns out Romney doesn’t oppose contraception at all – in fact, he thinks abortion should be an option in cases of rape, incest or to save a mother’s life.”

Joel Benenson, Obama’s pollster, said in an interview he did not believe Romney would succeed in changing minds, saying, “Perceptions are pretty baked in because of a range of positions on both the economic front and the social front.”

In a conference call with reporters, Cecile Richards, the president of Planned Parenthood, accused Romney of trying to “mislead the American people about his plan to turn women’s health care decisions over to their bosses.”

She pointed to Romney’s statement of support last winter for the so-called Blunt Amendment, which would allow employers to deny health coverage for procedures they find morally objectionable. (It came after he initially told an Ohio television station that he opposed the bill.)

No letup in sight

Obama campaign officials made it clear that they would continue to press the arguments about Romney’s record on women’s issues in advertisements, and they spent an additional $7 million on a final-stretch television blitz that was already costing $40 million.

Romney’s campaign had its own offensive ready. In a phone call with reporters, Sen. Kelly Ayotte of New Hampshire said “when the president talks about women’s issue he doesn’t talk about his vision for how he’s going to make sure that we address the number of women who are under- or unemployed.”

And late Wednesday, the Romney campaign released an advertisement featuring women who had served in his state administration who say they were struck by his “humanity,” say he “gets working women” and stands by single mothers.