Six Republicans bucked the party line Tuesday to open debate on a temporary reinstatement of lapsed jobless benefits — and for some, the story behind their support is one of high unemployment at home, combined with Democratic delegation-mates who create inherent political tension.

Jobless benefits expired on Dec. 28 for 1.3 million Americans, but states with higher unemployment rates are disproportionately affected because federal extended benefits are divided into tiers of aid based on need.

States that have unemployment rates at or above 6 percent, 7 percent and 9 percent have access to more federal benefits for their residents. Three of the GOP senators who voted with Democrats Tuesday — Dean Heller of Nevada, Dan Coats of Indiana and Rob Portman of Ohio — are from states where the unemployment rate is above 7 percent. They also all have Democratic seatmates, and had they not supported at least allowing debate of the measure, they risked headlines at home highlighting how they blocked assistance to their neediest constituents as their Democratic colleagues fought to approve them.

That was the case for Republicans Patrick J. Toomey of Pennsylvania and Mark S. Kirk of Illinois, both of whom fall into the same category as the aforementioned GOP supporters. The Philadelphia Inquirer ran the headline, “Toomey, Casey split on unemployment benefits” and the Chicago Tribune published, “Illinois senators split in jobless aid extension vote.” Pennsylvania and Illinois have jobless rates of 7.3 percent and 8.7 percent, respectively.

President Barack Obama also sought to pressure both House and Senate Republicans in a White House speech Tuesday.

But perhaps the biggest boon to Democrats seeking to approve the bill was that Republican leaders had not met with the rank and file to discuss voting strategy. When Coats was asked by CQ Roll Call about the decision-making behind his vote, the Indiana Republican said he had not spoken to any of his colleagues before voting.

“These are my own views. I can’t speak for the others. I just, frankly, think, this is one of those issues that goes directly to people who are hurting,” Coats said.

The Senate’s vote to advance the three-month extension, 60-37, came as a surprise even to leadership aides who had been counting the votes in the hours and days leading up to Tuesday. Democrats were so convinced the vote would fail that they already had given Sen. Mary L. Landrieu, D-La., the go-ahead to move forward with a news conference on a flood insurance bill they assumed would come to the floor next. And Senate Minority Leader Mitch McConnell, R-Ky., had offered before the vote to trade a one-year delay of the health care law’s individual mandate for the unemployment extension — a political maneuver he knew would never be palatable to Democrats but that belied a basic belief that he could hold his caucus together.

Of course, 60 votes also will be necessary to close debate on the legislation, which means Republicans have another chance to filibuster the bill later this week. With the exception of Heller, who co-sponsored the pending three-month extension with Democrat Jack Reed of Rhode Island, GOP senators are demanding offsets to pay for the bill. Democrats previously insisted they did not need to offset the cost of the legislation — given past reauthorizations of the program without pay-fors — but the pressure is now on them to change their position. By demonstrating a willingness to discuss solutions, the six Republicans who voted to open debate give GOP leaders unexpected cover to pin failure on Democrats.

“Why end the process from even starting? If Harry [Reid] wants to not give us an opportunity to not offer amendments, to make reforms, to accept the pay-fors, then Democrats will have to answer the question,” Coats told reporters after the vote. “Let’s force them to do it.”

Portman, in a statement after the vote, echoed that sentiment: “I voted to proceed with the debate over how to address unemployment insurance with the hope that during the debate the Senate will agree to pay for the extension and work to improve the unemployment insurance program so it works better to connect those unemployed with available jobs.”

It might not have been a coordinated strategy on the part of concerned Republicans, but it does appear to be working. Multiple Democratic aides said senators discussed the option of paying for the extension at their weekly caucus luncheon on Tuesday afternoon. And for the first time since Democrats began seriously discussing the expiration of unemployment benefits, a leading senator expressed an openness to exploring potential offsets. Reid on Tuesday talked about the possibility of using the more than $20 billion in projected savings from the pending farm bill — which aides expect to pass this month — to pay for the UI extension.

“The farm bill is less than $20 billion. I understand that they’ve cut some of the savings we had before,” Reid said. “Sometime this week the conference could be completed. I hope that’s the case. And certainly that’s a place we could look for some money.”

That idea has also been floated by some House Democrats.