WASHINGTON - Hours after President Obama signed sweeping health-care legislation into law Tuesday, the Senate began a debate on another piece of the package, giving Republicans one last chance to alter the bill before it begins to transform insurance coverage for millions of Americans. In the short term, GOP senators are aiming to gut the "fixes" package, a 150-page addendum to the new health-care law. Because the fixes bill was written under special budget reconciliation rules, it cannot be filibustered. But Republicans vowed to take full advantage of their right to offer unlimited amendments, intending to sabotage the package and create turmoil among Democrats who are counting on its passage. Republicans also launched a broader campaign to "repeal and replace" the new law, an initiative aimed at wooing voters in this fall's midterm elections. The pledge was quickly endorsed by GOP candidates including Carly Fiorina, who is challenging Sen. Barbara Boxer (D) in California, and Jane Norton, who is seeking to unseat Sen. Michael Bennet (D) in Colorado. "You're going to see this become, I think, one of the signature issues in the November 2010 election," said Sen. John Cornyn (Tex.), who is leading the drive to elect more GOP senators as chairman of the National Republican Senatorial Committee. Under reconciliation rules, the Senate will debate the fixes bill for 20 hours, then hold rapid-fire votes on amendments. That phase, being called a vote-a-rama, could begin as soon as Wednesday night and is likely to end sometime Friday, aides from both parties said, although Majority Leader Harry M. Reid (D-Nev.) has suggested that the Senate could finish Thursday night. Republicans have two goals this week: to use reconciliation rules to strike sections of the package and to offer amendments that prove impossible for Democrats to resist. If the fixes bill is changed in any way, it must return to the House for another vote. House members insisted on the bill as a condition of their passing the Senate's health-care bill, and Speaker Nancy Pelosi (D-Calif.) is expected to keep the chamber in session until Friday in case additional action is necessary. Congress is scheduled to recess for two weeks starting this weekend. Senate Democrats have scoured the bill for possible rules violations and say they see only a few minor areas of potential vulnerability. Budget Committee Chairman Kent Conrad (D-N.D.) said Tuesday that he is "highly confident" that the measure can survive even those challenges, adding: "We have spent an extraordinary amount of time going over this." Democrats claimed an important victory late Monday when Senate parliamentarian Alan Frumin rejected the most significant Republican challenge. Sen. Judd Gregg (N.H.) had argued that the measure should be stripped of its reconciliation status -- effectively killing the bill -- because provisions related to a new tax on high-cost health insurance policies would adversely impact the Social Security trust fund. Democrats countered that the impact is indirect, and in an e-mail to budget aides from both parties, Frumin said that the challenge "is not well taken." The reconciliation bill aims to change numerous provisions in the health-care law. Tax credits for people to buy insurance would be made somewhat more generous, particularly for lower-income Americans, and states would receive more federal funding to make Medicaid available to everyone who earns less than 133 percent of the federal poverty level. The fixes bill would offer a $250 rebate this year to seniors who fall into the "doughnut hole" coverage gap in the Medicare prescription drug benefit, and it would close that gap by 2020. It would cut an additional $60 billion from Medicare, bringing total cuts to the program to more than $500 billion over the next 10 years. And it would delay a tax on high-cost insurance polices until 2018, replacing the lost revenue by imposing the Medicare payroll tax on investment income for families earning more than $250,000 a year. As debate on the fixes unfolded on the Senate floor, both parties sought to capitalize politically on the passage of the health-care overhaul. Democrats circulated data from a new Gallup/USA Today poll that suggested public opinion on the legislation had started to shift in their favor.