Senate Majority Leader Harry Reid hoped to scramble enough support to pass an immigration reform bill during a second test vote expected Thursday evening, vowing to try to get it through the Senate despite a morning vote that failed to reach the 60-vote threshold needed to end debate and get to final passage.

Republicans stuck together to block an end to debate, 33-63. They opposed the measure because it would have prevented Republican amendments from being offered.

"It’s certainly not my goal to not get this bill to passage, provided we have fair treatment on this side of the aisle," Senate Minority Leader Mitch McConnell said.

"If we can't do this then we ought to vote to dissolve the Congress and go home till a new Congress," said Sen. Trent Lott, R-Miss., who added that he's getting calls from constituents upset that Congress can't pass an immigration reform bill.

Reid said he too did not want to give up on compromise legislation. "This is a bill that we have pushed it down the road a long ways, and I hope we can finish it," he said on the Senate floor before the morning vote.

Some senators are holding out hope that they can save the bill after it barely survived a deal-breaking challenge in an overnight vote. The legislation would address the 12 million illegal immigrants currently living in the United States, temporary workers and border security.

A FOX News poll finds that Americans think it would be better to enforce current immigration laws instead of an overhaul.

A 58 percent majority says the United States needs to enforce the current laws, while 34 percent think the country’s immigration laws need to be completely overhauled.

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The so-called "grand bargainers" who put the compromise legislation together met early Thursday morning for a last-ditch effort to pull it together. Sens. Edward Kennedy, D-Mass., and Jon Kyl, R-Ariz., aim to craft a compromise after the Senate voted early Thursday to sunset the temporary worker program after five years.

Kennedy allowed the amendment by Sen. Byron Dorgan, D-N.D., which had already been defeated once, to come up again for debate on the Senate floor after a minor change and some arm-twisting to convince one fellow Democrat to change his vote and support of the provision.

The Senate voted 49-48 to pass Dorgan's amendment.

Republican supporters see the temporary worker program, supported by business groups, as crucial to keeping the deal alive. Democratic critics like Dorgan said an open-ended program would drive down American workers' wages or take jobs away from Americans.

Aides say negotiators are working to alter the Dorgan amendment to satisfy both sides. Sen. Arlen Specter, R-Pa., said he or his allies would slightly re-word Dorgan's amendment and hope for a change of heart by one or more senators who "don't want to kill the bill."

Sen. John Cornyn, R-Texas, a vocal opponent of the legislation said the proposed temporary guest worker program only delays the continuing influx of illegals because immigrants will continue to arrive illegally after their short-term Z-visas expire following two-year work terms.

"This bill fails the essential test of our criminal judgment system. When we're looking at what the appropriate punishment is for people who violated laws, one of the goals is to deter future individuals from violating those laws. Instead of deterrence, this bill provides a huge magnet," he said.

The bill, backed by President Bush, survived many challenges on Wednesday.

One amendment would have postponed the bill's shift to an emphasis on education and skills among visa applicants as opposed to family connections. The other, offered by Sen. Barack Obama, D-Ill., would have ended a new point system for those seeking permanent resident "green cards" after five years rather than 14 years.

Along with the Dorgan amendment, the other two were seen as potentially fatal blows to the fragile coalition backing the bill, which remains under attack from the right and left. The bill would tighten borders, hike penalties for those who hire illegals and give many of the country's estimated 12 million illegal immigrants a pathway to legal status.

While the Dorgan amendment marked the biggest setback for the bill's advocates, there were others. They failed to defeat a Republican proposal to give law enforcement agents access to rejected visa applications, which could lead to the arrest and deportation of some illegal immigrants who otherwise might escape detection.

They also failed, by a 64-33 vote, to block a provision by Sen. James Inhofe, R-Okla., making English "the national language." Opponents called the measure demeaning and said they would try to kill it during House-Senate negotiations.

The Senate voted 51-46 to reject a proposal by Sen. John Cornyn, R-Texas, to bar criminals — including those ordered by judges to be deported — from gaining legal status. Democrats siphoned support from Cornyn's proposal by winning adoption of a rival version that would bar a more limited set of criminals, including certain gang members and sex offenders, from gaining legalization. The Senate backed that amendment 66-32.

The Senate also rejected a proposal by Sen. Robert Menendez, D-N.J., that bill supporters called a "killer amendment." It would have delayed the bill's shift in favor of attracting foreign workers with needed skills as opposed to keeping families together. Menendez won 53 votes, seven short of the 60 needed under a Senate procedural rule invoked by his opponents.

Menendez's proposal would have allowed more than 800,000 people who had applied for permanent legal status by the beginning of 2007 to obtain green cards based purely on their family connections — a preference the bill ends for most relatives who got in line after May 2005.

Kyl, a chief advocate of the bill, said most of the visa applicants Menendez wanted to help are so far back in line that it would be decades before the Homeland Security Department could process them. The Senate adopted Kyl's alternative, which would retain the family preference status for applicants who might win approval by 2026 under the department's projections.

Menendez, whose parents were Cuban immigrants, called the Kyl amendment "a fig leaf" that would make no meaningful change to the bill.

Presidential contenders featured prominently in the day's debates. Sen. Hillary R. Clinton, D-N.Y., fell short in her bid to remove limits on visas for the spouses and minor children of immigrants with permanent resident status.

Obama called the green card point system a risky "experiment in social engineering."

Cornyn had painted his criminals amendment as a "defining issue" for any presidential candidate — a sign of the degree to which the contentious debate is bleeding over into the GOP campaign fray.

Sen. John McCain, R-Ariz., alone among his party's presidential aspirants in backing the immigration measure, opposed Cornyn's bid and backed the Democratic alternative offered by Kennedy.

McCain was joined in opposing the amendment by the Senate's four Democratic presidential hopefuls: Obama, Clinton, Joe Biden of Delaware and Chris Dodd of Connecticut.

Cornyn prevailed on another matter opposed by the grand bargainers, however. His amendment, adopted 57 to 39, would make it easier to locate and deport illegal immigrants whose visa applications are rejected.

The bill would have barred law enforcement agencies from seeing applications for so-called Z visas, which can lead to citizenship if granted. Cornyn said legal authorities should know if applicants have criminal records that would warrant their deportation.

Opponents said eligible applicants might be afraid to file applications if they believe they are connected to deportation actions. Sen. Dianne Feinstein, D-Calif., said in an interview that Cornyn's amendment was "not a deal-killer" but would have to be changed in House-Senate negotiations.

Other amendments defeated Wednesday included a Democratic effort to alter the temporary guest worker program that would be created by the bill.

Sen. Jeff Bingaman of New Mexico wanted to allow workers to come for six consecutive years. The Senate voted 57-41 to reject the amendment, retaining the bill's call for most guest workers to go home for a year between each of three two-year stints.

The Senate also rejected an amendment by Sen. Jim DeMint, R-S.C., to change the Z visa program whereby illegal immigrants could gain lawful status. DeMint proposed requiring them to buy high-deductible health plans to be eligible for visas.

FOX News' Trish Turner and The Associated Press contributed to this report.