Updated

The Pentagon announced Wednesday that a charity will step in to pay death benefits to the families of fallen soldiers, in response to widespread outrage over the payments being cut off amid the partial government shutdown.

Earlier in the day, the head of the Veterans of Foreign Wars slammed the cut-off as "absolutely appalling."

The House voted Wednesday afternoon to restore the $100,000 payments. But in a rapid turn of events, the Pentagon announced minutes later that it had entered into an agreement with the non-profit Fisher House Foundation to keep the payments flowing to families -- without the need for congressional action. (Though the House approved its bill unanimously, senior Senate sources told Fox News that the Senate did not plan to take it up.)

"In consultation with the Office of Management and Budget, DoD has determined that we can enter into a contract with the Fisher House Foundation to provide these benefits," Defense Secretary Chuck Hagel said in a statement, while saying he was "offended" and "outraged" that the partial shutdown prevented the Pentagon from making the payments.

GOP leaders have disputed that point. They say legislation approved last week to pay the military should have given the Pentagon the ability to pay death benefits.

More On This...

    Hagel claimed the administration still "lacked the necessary authority," and so turned to the Fisher House Foundation for help.

    "After the shutdown ends, DoD will reimburse the Fisher House for the costs it has incurred," he said.

    William Thien, the VFW national commander, earlier in the day called the benefit cut-off the "last straw" in the effects of the partial shutdown.

    "It is absolutely appalling and nothing short of a travesty that elected officials continue to receive paychecks and benefits while not providing for those who deserve it most," he said.

    White House Press Secretary Jay Carney said President Obama was "very disturbed to learn of this problem" and directed the Pentagon to work with the White House budget office to correct it.

    The number of families affected is far greater than previously thought. It was initially confirmed that the families of five servicemembers, who were killed since the partial shutdown went into effect, had their $100,000 benefit payments delayed.

    A senior U.S. defense official told Fox News that the total number of families affected is actually 26 -- the number of U.S. servicemembers who have died since Oct. 1. Twenty of them died in the U.S.; six were killed in Afghanistan. All 26 families were not receiving the $100,000 payments.

    The treatment of veterans amid the partial shutdown has fueled anger and frustration. Veterans Affairs Secretary Eric Shinseki testified before a House committee on Wednesday that if the impasse is not resolved by the end of the month, millions of veterans will see their compensation and other VA payments halted. He said he cannot act without a budget.

    "This is totally unacceptable and disgraceful that our elected leaders in Washington would allow this to happen," Thien said in a written statement. "We need leadership, not more rhetoric, and if the government is unable to take care of veterans, then the government should quit creating us."

    The remains of four of the six soldiers killed in Afghanistan were returned on Wednesday to Dover Air Force Base, Del. Hagel traveled to the air force base for the arrival.

    The four soldiers killed Sunday are 1st Lt. Jennifer M. Moreno, 25, of San Diego, Calif.; Sgt. Patrick C. Hawkins, 25, of Carlisle, Pa.; Sgt. Joseph M. Peters, 24, of Springfield, Mo., and Pfc. Cody J. Patterson, 24, of Philomath, Ore.

    Rep. Duncan Hunter, R-Calif., wrote to Hagel on Tuesday complaining about the suspension of the $100,000 death benefit and accusing the Pentagon of making that decision on the basis of a "careless legal interpretation."

    Administration officials have said the decision was based on a legal review by government lawyers, including the Justice Department.

    The Associated Press contributed to this report.