WASHINGTON (Reuters) - An attempt by U.S. Senate Democrats to override the Supreme Court's controversial birth control ruling failed to muster enough votes to move forward on Wednesday, but lawmakers vowed to keep pressing the issue heading into the midterm elections.

Senators, including three Republicans, voted 56-43 for the bill, which would bar employers from discriminating against female employees in coverage of preventive health services, including contraception.

That was short of the 60-vote hurdle needed to move the bill forward. But Senate Democratic leaders promised to bring the contraception issue up again.

"Women across the country today watched as all but three Republicans showed they care more about protecting the rights of CEOs and corporations than about protecting the rights of women to access critical healthcare coverage," Senator Patty Murray of Washington, one of the bill's sponsors, said at a news conference.

The Supreme Court's 5-to-4 decision on June 30 allowed corporations held by a family or a small number of people to forgo for religious reasons requirements in President Barack Obama's signature healthcare reform law that employers' health plans cover birth control.

The court's male justices sided with retail arts and crafts supplier Hobby Lobby, whose owners objected to providing employees certain methods of contraception that the owners considered forms of abortion.

Murray and Senator Mark Udall of Colorado responded last week with the bill to override that decision. Forty-four other Democrats signed on as co-sponsors, and it was endorsed by medical groups such as the American Congress of Obstetricians and Gynecologists.

Democrats in the House of Representatives have a companion bill that stands little chance of passage in the Republican-controlled body.

"We are going to keep working on this until we get it done," Murray told Reuters on Tuesday. "This is the right thing to do. We're going to fix this policy."

"This is far from over," Udall said after the vote.

Senate Majority Leader Harry Reid said the Senate would vote on the issue again this year.

MIDTERM ISSUE

Political observers said all along that Senate Democrats could not garner the Republican support needed to reach 60 votes. Instead, they had their eye on November elections.

"This will be a huge motivator for women in the fall and a liability for Republican candidates up and down the map," said Justin Barasky, a spokesman for the Democratic Senatorial Campaign Committee.

Roughly one-third of the seats in the Senate and the entire House of Representatives are in play. Democrats face a tough battle to hang onto their majority in the Senate.

In 2012, Democrats hung onto some seats by painting Republicans as anti-women, after several candidates made remarks about rape and other issues that were seen as insensitive.

North Carolina Senator Kay Hagan, a Democrat who faces one of the toughest re-election contests this year against Republican state lawmaker Thom Tillis, has made the issue a key part of her campaign, spokeswoman Sadie Weiner said.

"Kay's excited to know that women are backing her and are spending time volunteering, knocking on doors," Weiner said. "It's certainly drawn attention and put it on the radar of North Carolina women that there is a contrast in this race when it comes to access to contraception."

Senate Republicans have announced their own post-Hobby Lobby bill to ensure employers could not block their employees from obtaining birth control.

Republican Senators Lamar Alexander of Tennessee and Orrin Hatch of Utah said before Wednesday's vote that the Hobby Lobby ruling was about constitutional religious freedoms, not women's rights.

Hatch told Reuters he was not worried the bill would persuade women to vote for Democrats in November.

"I think there are a lot of very intelligent women out there who know this is a political game, and it shouldn't have an effect," he said.