WASHINGTON — More than a month after a frenzied U.S. effort to evacuate thousands facing retribution from the Taliban in Afghanistan, members of Congress are still quietly pushing the government to help extract a small group of stranded Afghans who are direct relatives of American military service members.

The service members, some of whom have traveled to Washington to plead with lawmakers and the Biden administration for help, largely share the same story. Many of them once worked as interpreters or fixers for the U.S. military in Afghanistan but moved to the United States years ago, obtaining visas and then green cards to become permanent residents, and then enlisting in the armed forces they had once served as civilians.

They were evacuated from Afghanistan as part of the U.S. withdrawal weeks ago. But now, with the Taliban seeking to punish anyone with ties to the Americans, their parents and siblings are in danger, and lawmakers and U.S. officials are puzzling over how to help them.

“It was bad enough that American citizens were left behind, and our Afghan partners were left behind,” Representative Michael McCaul of Texas, the top Republican on the Foreign Affairs Committee, said in a recent interview. “But when I found out that we have active service members whose families are stuck in Afghanistan, and the State Department can’t get them out? It was like the worst of the worst cases.”

The cluster of service members’ relatives, estimated in the low hundreds, is one of several groups of at-risk people whom lawmakers are still working behind the scenes to help, underscoring the danger many Afghans still face in their home country.

The State Department in recent weeks created a team across agencies, including the Defense Department, to help facilitate the evacuation of service members’ families and other vulnerable Afghans, according to a spokesman. But officials have provided little information about how that will be accomplished and who will be eligible.

After Kabul fell to the Taliban in August, congressional offices were flooded with messages asking for help and staff sprang into action, morphing the offices into informal operations centers focused on assisting Americans and Afghans escape as the last U.S. troops prepared to withdraw. Two months later, lawmakers are still agonizing over the steady stream of requests they continue to receive.

“We need to get them out because the Taliban is actively hunting,” said Representative Jason Crow, a Democrat of Colorado and a former Army Ranger who served in Afghanistan. “They are systematically working through their lists. Time is of the essence here.”

Lt. Gen. Douglas E. Lute, who directed Afghan strategy at the National Security Council for Presidents George W. Bush and Barack Obama and is now retired, testified before the House Foreign Affairs Committee last week that he was aware of at least 35 service members who were still trying to evacuate their family members from Afghanistan. Lawmakers tracking the issue have similar estimates.

Lawmakers first learned of the relatives’ plight in August, during a news conference held by Mr. McCaul outside the Capitol. As he was delivering remarks on the situation of stranded Afghan interpreters, he noticed a group of men in Army uniforms holding placards.

“Our families should not get slaughtered as a result of our service in the U.S. military,” one sign read.

Their precarious situation reflects the limitations of a special visa program that was already badly backlogged and never meant to facilitate a mass evacuation. The Special Immigrant Visa program was intended to help people facing threats because of their work for the United States government — a definition that excludes many of those whom the Taliban is now targeting.

“The S.I.V. program has been so fundamentally broken for so long, there are so many aspects that need to be fixed,” Mr. Crow said.

Last week, he said he and his family had welcomed an Afghan family that his office had helped to evacuate in August.

“They had been waiting on S.I.V. approval since 2005 and they probably would still be waiting,” he said.

Over the summer, Congress passed legislation to increase the cap on the special visas by 8,000 and to remove application requirements that slowed down the process. President Biden signed that bill into law, but the program’s limitations remain, and now Mr. Crow and Representative Peter Meijer, Republican of Michigan, are proposing a measure to raise the cap by 10,000 more and expand eligibility.

But lawmakers also have complained that the visa program’s bureaucratic issues have only been compounded by the way in which a strained State Department has handled their requests.

State Department officials have designated only American citizens and their immediate family members as the top priority for evacuation, according to congressional offices.

Mr. Crow said his office was looking for better guidance from the administration “on how folks can be evacuated and who is eligible for various programs.” Mr. McCaul said that Antony J. Blinken, the secretary of state, had never responded to a letter he had sent inquiring how many service members had family currently trapped in Afghanistan and what the agency was doing to secure their evacuation.

“If you’re a U.S. citizen with your blue passport, I think there’s hope for you,” Representative Michael Waltz, Republican of Florida, said in an interview. “But we are seeing a marked difference across all the groups we’re dealing with, which is really disturbing.”

Mr. Waltz said he understood the State Department’s dilemma.

“Where do you draw that line: nuclear family, brothers, cousins, parents?” he said. “But the point is, I think we should have had as much latitude as possible.”

The State Department spokesman said efforts were underway to try to help the service members whose Afghan family members were in need. But he conceded that the department would always have a “special responsibility” to Americans seeking evacuation.

Amplifying the sense of urgency, lawmakers and veterans’ groups working on evacuation efforts said, is an uptick in retributive attacks by the Taliban. Mr. Waltz said that his office had recently received video of a woman who had served in the Afghan National Army who had been nearly beaten to death, her arms and ribs broken.

After American military forces left Kabul in August, Daniel Elkins, the executive director of the Special Operations Association of America, a veterans’ organization that has helped organize evacuation efforts, said he saw a downturn in Afghans contacting him for help.

“Recently there’s been a steady increase in number of people reaching out again,” Mr. Elkins said. “That communicates that the threat picture has been rising.”

Lawmakers have also taken note of another vulnerable group who are ineligible for the special visa program: the elite commandos who served in the Afghan Special Forces, who were trained and equipped by U.S. forces and often worked alongside them.

Mr. McCaul said he raised the issue on a call on Wednesday with Wendy Sherman, the deputy secretary of state, who assured him the State Department was working on it.

But, Mr. McCaul added, “it’s hard to process” any applications “when you don’t have an embassy in country.”

Heather Nauert, a former State Department spokeswoman during the Trump administration who has been deeply involved with advocating for the evacuation of Afghans and American service members’ families, said that the commandos she has spoken to have reported an increase in violence targeting them and their families.

“Every day, they are getting images and stories from their former comrades stuck in Afghanistan,” Ms. Nauert said. “Some of them have had family members who have been hunted captured, tortured and murdered.

“These are men who were trained by America’s special operators,” she added. “They never applied for visas, because they never thought they would have to.”