California has had a gold rush of people gaining health coverage—but the state will face a tougher time mining the remaining uninsured during Obamacare's next open-enrollment season, a new study suggests.
The Golden State saw about 3.4 million previously uninsured adults obtaining health insurance since last fall's launch of Obamacare. That equals 58 percent of the total uninsured population as of last year, according to the Kaiser Family Foundation survey.
The survey found that the bulk of those still lacking insurance are Hispanics. And many of them who are eligible for Obamacare and Medicaid are reluctant to enroll because they fear it will bring attention to their family's immigration status.
The big gains in reducing California's uninsured population came after the state enthusiastically bought into the Affordable Care Act by expanding eligibility for its Medicaid program, setting up its own health insurance exchange and aggressively marketing both options.
Overall, California had 1.7 million people deemed eligible for its Medicaid program, and 1.4 million others signed up for Obamacare plans as of mid-April.
While the state has about 12 percent of the total U.S. population, its Medicaid tally was 25 percent of the total U.S. eligibility determinations, and its Obamacare tally was 17.5 percent of nationwide signups.
"I do think that California could be a model for states if they did choose to embrace the ACA as much as California did," said Mollyann Brodie, who oversees the foundation's public opinion surveys.
Read MoreSubsidy, what subsidy? Obamacare's clueless
The survey, released Wednesday, is the second in a series that aims to track the experiences of 2,001 uninsured state residents.
Most of the reduction in the numbers of uninsured Californians came from the two programs that are key to ACA's goal of getting all Americans some form of health coverage, the survey found.
Medi-Cal, the state's Medicaid program for the poor, signed up 25 percent of the state's uninsured population. (California is one of 27 states that have expanded or intended to expand eligibility for its Medicaid program to include nearly all poor adults. Those states have tended to see marked gains in their Medicaid rolls not only from the newly eligible, but also from the previously eligible.)
Covered California, the government-run exchange that sells insurance plans offered by private companies, signed up 9 percent of the total uninsured, the survey found.
The remaining gains came from employer-provided insurance, which signed up 12 percent of the previously uninsured, and individual insurance sold outside the Covered California exchange, which signed up 5 percent of the previously uninsured.
Eight percent were insured through other sources.
The survey found similar results in the percentages of people in different ethnic groups who gained insurance. Sixty-one percent of previously uninsured whites gained coverage, 62 percent for blacks and 52 percent for Hispanics.
Brodie said the survey found that most people who signed up for either Medi-Cal or Obamare plans did so after receiving some kind assistance navigating the enrollment application.
"Six in 10 of them told us they needed someone to actually help them through the process," Brodie said.
Of those newly insured, 73 percent said their plans are a good value. But 46 percent of the newly insured in plans other than Medi-Cal say they find that paying for their coverage is difficult.
Despite California's relative success in getting coverage to many previously uninsured people, the state has its work cut out for it in getting the rest covered, the survey found.
"Expanding coverage gets harder from here," said Kaiser Family Foundation President and CEO Drew Altman.
Kaiser noted its survey found that of the 42 percent who remain uninsured, "many have characteristics that traditionally make them hard to reach with health coverage."
Read MoreThere may be a cheap fix for an Obamacare quandary
Those include the fact that many of those people—37 percent—have never had health insurance, and 45 percent have been uninsured for at least two years.
Slightly more than a third of the remaining uninsured said they tried to get coverage but didn't follow through because they believed it was too expensive or had a tough time completing the process.
Hispanics make up 62 percent of the uninsured, presenting a particular challenge for insurance-expansion advocates.
"They've always been a majority of the uninsured" in California, Brodie said.
Nearly half of the uninsured Hispanics are ineligible for either Medi-Cal or Obamacare plans because they are undocumented immigrants.
But even Hispanics who are in the country legally, and who are eligible for those programs, may be less apt to sign up for coverage than other groups.
Thirty-seven percent of those eligible, documented Hispanics told researchers they were "very worried" or "somewhat worried" that if they signed up for health insurance it "will draw attention" to their own or a family member's immigration status.
That's despite the fact the federal immigration authorities have said insurance application information will not be used to enforce immigration laws.
"One of the biggest challenges going forward is a big share of the remaining uninsured have these fears or concerns that may create an additional barrier to getting them insured in the next open enrollment season," Brodie said.
But, she noted, there are incentives for the uninsured to sign up.
Brodie said that about 40 percent of the remaining uninsured in California should be eligible for the Medi-Cal program, which is free for enrollees. Twenty-five percent should be eligible for federal, taxpayer-funded financial aid or subsidies to help offset prices for the Covered California exchange.
Larry Hicks, a spokesman for the exchange, said, "We're always looking for better ways to improve outreach."
Read MoreObamacare could be saved by 'nuclear option'
"We hope that our message will be a little sharper" for the next open enrollment in Obamacare plans, which starts in mid-November, Hicks said.
The exchange plans to award nearly $17 million in grants for its "Navigator Program," which will educate consumers about options and financial aid. The goal is to enroll more than 130,000 people during the next open enrollment.
"We are trying to develop a culture of coverage among Californians," Hicks said. "We didn't expect everyone would have insurance in the first year, and we would be pleasantly surprised if that happens in the second year. ... It's a process."
—By CNBC's Dan Mangan