The good news for trade supporters is that House Republicans just bought themselves more time to complete their agenda. The bad news is they still don't know what to do with it.

The House punted on a stalled trade deal Monday, leaving President Obama's key domestic-policy priority unresolved, just days after the chamber shot down a key piece of the deal.

The House Rules Committee approved a rule—which still must be passed by the full chamber—allowing the House until July 30 to reconsider Trade Adjustment Assistance, an aid and job-training program that failed last week when anti-trade Democrats voted with Republicans who do not like government aid programs to reject it. The measure is seen as a crucial component to passing Trade Promotion Authority, a fast-track mechanism that would grease the wheels for Congress to pass the sweeping, secretive Pacific Rim trade deal later this year.

House Speaker John Boehner spoke with Obama by phone Monday, although neither office revealed details of the call. Nevertheless, it is clear that Boehner and his leadership team want the administration to work harder to draw Democrats to the table.

"We remain committed to getting TPA done, and this will give the president more time to communicate the consequences of not moving forward with his party," said Boehner spokesman Kevin Smith.

Earlier in the day, however, the White House placed the onus squarely on Congress. Obama spokesman Josh Earnest played down the consequences of Friday's vote, blaming the holdup instead on procedural issues.

"The president and the rest of us here at the White House continue to be confident that there is strong bipartisan support for this approach," Earnest told reporters. "We just have to figure out how to untangle the legislative snafu in the House."

The House's pro-trade Democrats are regrouping as well. Rep. Henry Cuellar said the 28 Democrats who voted in favor of TPA are meeting with the White House at 8:30 a.m. Tuesday to talk strategy.

While trade backers have yet to rally around a Plan B to move TPA forward, Cuellar conceded that the TAA revote probably won't be the solution. "There might be some other ways to look at this," he said, adding that the path forward would become clearer once Republican leaders assess the numbers in their caucus.

Rules Committee ranking member Louise Slaughter, discussing pushing back the TAA vote, also said it would be unusual to see many members change their vote the second time around.

The TAA measure is seen as critical in the Senate, because several senators who voted in favor of the TPA measure did so only because it was combined with TAA. But without TAA, Republican Sen. John Thune conceded, the math gets more difficult.

"Trade got a lot more complicated last week," he said. "I'm sure the House is going back to the drawing board, and I don't know that they have a solution yet. But obviously if they can figure out a way to get TPA back over here, then we're going to have to figure out a way to get TPA and TAA done—but probably in independent paths."

GOP Sen. Rob Portman was also skeptical that a stand-alone TPA bill could get the 60 votes needed to advance without the TAA component. "I don't know that, but I think it would be difficult," he said.

Portman faces a tough reelection race in Ohio, a state with a significant number of displaced manufacturing workers. Asked whether he would support a clean TPA measure, Portman said, "Hopefully I won't have to make that decision."

Meanwhile, the pressure from off the Hill has yet to relent either. Organized labor has invested considerable resources to kill TPA, and groups like the AFL-CIO and United Steelworkers are urging members to make sure the revote meets the same fate as last week's attempt. The AFL-CIO launched online and in-district ads thanking the members who voted against the bill.

Alex Rogers and Rebecca Nelson contributed to this article.