Sen. Ted Cruz (R-Tex.) speaks to reporters Wednesday on Capitol Hill. (Melina Mara/The Washington Post)

Fresh off an unsuccessful attempt to block the president’s sweeping Affordable Care Act, several conservative Republicans announced Thursday that they have decided on their next political target: the Affordable Care Act.

The temporary resolution of the budget battle is likely to intensify, rather than lessen, public scrutiny of the health-care law, commonly known as Obamacare. Chronic problems with the online enrollment system — which have diminished but not disappeared since its Oct. 1 launch — were largely overshadowed by the 16-day fiscal standoff in Washington.

But several GOP senators said Thursday that they will work to shine a spotlight on the problems and renew efforts to stop the law, although they did not specify how.

“I would do anything, and I will continue to do anything, to stop the train wreck that is Obamacare,” said Sen. Ted Cruz (Tex.), a tea party favorite who said he would not rule out pushing for another government shutdown.

House Energy and Commerce Committee leaders announced Thursday that they will hold a hearing Oct. 24 to scrutinize the law’s implementation. They also sent Health and Human Services Secretary Kathleen Sebelius a letter, asking her to reconsider the administration’s decision not to participate.

Key moments from President Obama's speech the morning after Congress passed a deal to raise the debt ceiling and reopen the government. (The Washington Post)

Sen. David Vitter (R-La.), who tried unsuccessfully to insert a provision in Wednesday’s continuing resolution that would have denied federal health insurance subsidies to lawmakers and their aides, said he will revive the proposal as soon as possible.

“I guarantee it will be back,” Vitter said on Fox News. “I’m not going away, and this issue is certainly not going away.”

More broadly, Republicans have begun to reposition the issue as a referendum in the 2014 midterm elections. Speaking on Fox News’s “Hannity,” Sen. Marco Rubio (Fla.) said Wednesday night that Republicans will make a strong case against the health-care law next year when “the realities of the law are going to begin to impact people.”

“There is going to be an all-out revolt in this country over that,” he said. “And that is, I think, the moment to absolutely act and say we are going to get rid of this law and then look for opportunities in the future to replace it.”

Matt Kibbe, president of the FreedomWorks conservative group, said several Democratic senators up for reelection in red states will have trouble defending the program given its current troubles.

“They’re not going to want to explain the failures of the system,” said Kibbe, whose group plans to hold rallies this month with young people who oppose the legislation’s individual mandate. “If you’re forcing people under penalty of law to sign up for something you can’t sign up for, that’s a problem.”

Health and Human Services spokeswoman Joanne Peters said officials “continue to work around the clock to improve the consumer experience on,” the federally run Web site, adding that 17 million unique visitors came to the site in the first two weeks.

Peters would not specify how many consumers were creating accounts or enrolling in coverage, saying the agency would release those numbers next month.

One administration official, who spoke on the condition of anonymity to discuss operations, said a team of federal officials and contractors is working on the site. Most of the contracts “included maintenance and fixes,” the official added, so the government was not incurring a major additional cost. Another official, who spoke on the condition of anonymity for the same reason, said wait times at telephone call-in centers had shrunk to two seconds.

David Simas, assistant to the president, wrote in an e-mail that Republicans are making a mistake if they continue to attack the health-care law. Overall favorability ratings for the Republican Party have dropped to historic lows, according to a recent NBC/Wall Street Journal poll, while the health-care law’s popularity rose seven points in the same survey.

“Bottom line, if they believe that running the same play and repealing the Affordable Care Act is a winner in 2014, they, once again, are badly misreading the American people who want us to move on,” he wrote.

Even some of the law’s strongest backers, such as House Minority Leader Nancy Pelosi (D-Calif.), said the administration has limited time to fix and the enrollment process.

“They were overwhelmed by the traffic, okay. But now let’s see how long it’s going to take to have that be fixed,” Pelosi told reporters Thursday. “I hope that we would have some answers soon and that the answer would be, okay, we found the glitch or whatever it is, it’s been corrected, and here is a demonstration as to how people, when they approach it now, will be received.”

Families USA President Ron Pollack, who supports the law and has been participating on twice-daily calls with Health and Human Services officials, said there is an “enormous amount of time and staffing being devoted to getting the problems fixed.” He said the site’s glitches will pose a serious problem only if they extend into next month, when a larger number of people will seek to enroll by Dec. 15 to begin coverage in January.

“That means if we haven’t seen very substantial progress before the end of this month, that’s worrisome,” Pollack said.

In the meantime, officials charged with getting people enrolled are focused on discussing options with uninsured Americans rather than trying to sign them up on the federal marketplace.

Cathy Levine, who directs the Universal Health Care Action Network of Ohio and is a certified application counselor, said she and her colleagues have been holding public meetings and telethons to publicize the law but scheduling enrollment appointments for next month.

“What’s happening on the ground is conversations are beginning between people who have been uninsured for a long time and assistors about what their new coverage options are,” said Levine, who met with Sebelius in Columbus, Ohio, on Thursday. “When people express frustration to us, we answer that they will have plenty of time and opportunity to enroll by December 15.”

While the federally run site has stumbled, the Obama administration has touted the success of state marketplaces, which have been running more smoothly. “Great news: The number of Oregonians without health insurance dropped 10% in 2 weeks,” the White House tweeted Thursday. “#Thanks­Obamacare #GetCovered.”

Sarah Kliff contributed to this report.