The failure of those troubled exchanges to enroll more than a relatively few people in Obamacare plans for nearly two months lead to widespread speculation that Democrats would get pummeled in the mid-term 2014 elections, because the health reform law was crafted by a Democratic president and passed by what was then a Democratic-dominated House and Senate.
But after aggressive repair effort largely fixed HealthCare.gov by the winter, enrollment efforts dramatically rebounded, ending with 8 million selecting an Obamacare plan by mid-April.
While Republican leaders have remained critical of the law and its mandate that nearly all Americans obtain health insurance or pay a fine, there was a pullback in rhetoric about repealing the law after the close of open enrollment. And some Democratic candidates have been using Obamacare as a talking point in their races.
When Bankrate.com last asked people whether Obamacare should be repealed or kept as-is, 45 percent of respondents said it should be junked. But in the new poll, just 30 percent of respondents said the new Congress should repeal the new law.
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A total of 52 percent of people want the next Congress to make either minor or major changes to the law. Just 12 percent of the people said the law should remain the same as it is now.
This was the first time that Bankrate in its poll asked people whether they believed the law should be kept, but modified. Beforehand, the question was whether it should be repealed or kept as-is, Whiteman said.
"I think it's interesting that when you give "change" as a choice, that comes out on the top," Whiteman said.
Whiteman also noted that the number of people who told the poll that their health insurance situation had improved was double from when that question was asked almost a year ago.
In August 2013, 8 percent of respondents said they had seen improvement in their health insurance situation. By this June, that had doubled to 16 percent, he noted.
—By CNBC's Dan Mangan