Senate Majority Leader Harry Reid obtained the surprise delay on Monday just as the chamber was expected to begin a roll-call vote on whether to start consideration of the measure.
With Congress returning for the new year, it had been unclear if the bill would muster the needed 60 votes in the 100-member Senate to end a Republican procedural roadblock.
But with much of the nation in the grips of a deep freeze, more than a dozen lawmakers were prevented from even getting to the Senate, making it certain that backers would fall short.
Democrats hold the Senate, 55-45.
Assistant Senate Republican Leader John Cornyn of Texas accused Reid of pushing ahead with a vote because he was more interested in the politics of the issue than helping the jobless.
"This ought to be postponed to a later time when we could have a real debate" about how to "pay for an extension of jobless benefits and how to get the economy growing," Cornyn said.
Reid then rose and, without objection, got the vote reset for 10 a.m. on Tuesday. A Senate aide said Democrats believe they are within one or two votes of getting 60.
If Democrats fall short, they may agree to Republicans demands for some sort of spending cuts to cover the cost of extending the program without increasing the record federal debt, aides said.
At an estimated cost of $6 billion, the bill offered by Democratic Senator Jack Reed of Rhode Island and Republican Senator Dean Heller of Nevada would immediately extend for three months the Emergency Unemployment Compensation program.
During those three months, backers say, Democrats and Republicans could consider possible ways to offset the cost and create new jobs.
Signed into law in 2008 by Republican President George W. Bush, the program last year provided the jobless an average of $300 per week for an additional 28 weeks once state benefits ended.
About 1.3 million Americans lost their benefits on December 28. Unless the program is renewed, another 2 million are expected to lose their benefits in the first six months of this year.
"Don't leave these people without anything," Reed pleaded on the Senate floor.
Supporters argue that besides helping the unemployed, the program boosts the economy as recipients quickly spend their benefit checks on essential goods, helping local retailers.
Reed said if backers of his bill fail to muster the 60 votes needed to advance the measure, they would keep pushing to bolster support in coming days.
"Public pressure will build," Reed said. "People need help."
Obama intends to help whip up such public support with an event at the White House on Tuesday with long-term unemployed.
Republicans contend that the U.S. economy, with the jobless rate now at a five-year low of 7 percent, is on the mend and that such emergency federal assistance is no longer necessary.
The best way to spread the wealth, Republicans argue, is to create more of it by creating jobs via less regulation and taxes.
While Heller urged lawmakers to support the bill, he also echoed a position of fellow Republicans, saying, "Growing our economy should be the primary focus and concern of this body."
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