After they get the website fixed, then what? Keeping your doctors and hospitals may be the next vexing challenge for Americans in the new health plans created by President Barack Obama’s law.

Obama promised people could keep their doctors. But in many states the new plans appear to offer a narrow choice of hospitals and doctors. Overall, it’s shaping up as less choice than what people get through Medicare or employer-based coverage. Also, it can get complicated tracking down which medical providers are in what plans.

“The next shoe is going to drop sometime after Jan. 1, when people actually start using their plans,” said health economist Gail Wilensky, who ran Medicare for President George H.W. Bush. “Whether or not they can keep their doctor is going to depend on whether their doctor was chosen — or wanted to be — part of a plan on the (insurance) exchange.”

Narrow networks are part of the economic trade-off for keeping premiums under control in Obama’s health insurance markets, the new gateway to coverage for people who don’t have job-based plans. Technical problems with the website HealthCare.gov have dampened initial signups, but 7 million people are expected to participate in the insurance exchanges next year. Some of those people already have coverage through individual plans.

Exchange plans are required to take all applicants, cover broad benefits and provide robust financial protection against catastrophic illness. In return for that, something else has to give. The result: limited choices and significant out-of-pocket costs through deductibles and copayments.

“To get that low premium, the way to get there is by having a more limited or narrow provider network,” said Matthew Eyles, a vice president of Avalere Health, a market analysis firm that is closely following developments.

The Obama administration says none of that takes away from what will still be a historic improvement for uninsured people. Exchanges “will vastly increase uninsured Americans’ access to providers, giving them an alternative to the emergency room or community health centers,” said Health and Human Services Department spokeswoman Joanne Peters. The health care law sets standards for insurers to provide adequate networks, she added.

But the American Medical Association has questions.

“Although it may be too early to reach any definitive conclusions, we are monitoring the adequacy of the exchange networks and will be analyzing the impact of these restrictive strategies on patient access to care,” said Dr. Ardis Dee Hoven, the organization’s president.

In some states, it’s already recognized as a problem. It’s magnified in New Hampshire because just one insurer, Anthem Blue Cross and Blue Shield, is offering plans there through the new marketplace. The company’s new network for individual plans excludes 10 of the state’s 26 frontline hospitals.

That was a rude awakening for Dr. Jocelyn Caple, who works at one of the excluded hospitals, Frisbie Memorial in Rochester, N.H., where she’s president of the medical staff. She and her family are currently covered under their own policy with Anthem.

“We all use Frisbie physicians,” she said. Although there’s another facility she could go to next year, “it’s pretty shocking not be able to come to your own hospital.”

As an Obama supporter, Caple is troubled. The hospital is considering a lawsuit.

All these years, Frisbie has performed a balancing act caring for a mix of insured and uninsured patients, she said. “Now at a time when you would have thought that the coming of insurance would make it easier to care for those patients, we are being denied the opportunity to care for them,” she added. “It’s pretty disturbing.”

Narrow networks also are raising concerns in other states, and hospitals appear to be the first squeaky wheel.

In Kentucky, where three companies are selling insurance through the state exchange, six hospitals have lodged complaints with the insurance department regarding the network issue. In Maine, Anthem Blue Cross and Blue Shield’s proposal to exclude six hospitals in the southern part of the state has been heavily criticized. The state partially approved the proposal.

In Chicago, Rush University Medical Center used to be covered “in network” by all the health plans offered in Cook County, Illinois’ largest county. Now, of the 71 plans serving the county on the state’s exchange, Rush is covered by only 38 plans.

In New York, one of the world’s foremost cancer hospitals, the Memorial Sloan-Kettering Cancer Center, is not “in network” for any of the insurance plans on the state’s exchange.

Insurers have used limits on choice to control costs for years. Avalere analyst Eyles said the health care law’s narrow networks may gain acceptance if they are able to deliver timely, high-quality care at lower cost. If that’s the case, bet on employers to experiment with them.