More than 700,000 employees face unpaid leave due to the shutdown which was triggered after the two houses of Congress did not agree on a new budget.
Hyundai said affected employees who currently own its vehicles will be given a payment relief "for as long as they are out of work".
Employees looking to buy a new car will be given a 90-day payment deferral.
"We recognize the impact on family budgets that the furlough will drive," John Krafcik, chief executive of Hyundai Motor America, said in a statement.
Hyundai had offered a similar scheme, the Hyundai Assurance programme, during the peak of the global financial crisis four years ago to help consumers who had lost their jobs.
Many analysts have said that the move had helped the South Korean firm win customer loyalty and boosted its sales in recent years.
The company said that its latest offer to help the federal employees was an addition to that programme and aimed at "helping workers at a time when they most need it".
"Like we did almost four years ago when we launched Hyundai Assurance, this is our way of saying 'We've got your back' during this uncertain time," Mr Krafcik said.
Under the latest offer, Hyundai will extend all auto loan and lease payments during the shutdown for current Hyundai owners who are put on unpaid leave.
The programme is available to all customers who have financed their purchase or lease through Hyundai Finance America.

The Labour leadership frontrunner made the remarks in 2011, shortly after the al-Qaeda chief was shot dead.
A spokesman for Mr Corbyn defended the remarks saying he was "a total opponent of al-Qaeda, all it stands for".
Meanwhile Andy Burnham appealed to undecided voters, saying it was still "all to play for" in the election.
The other candidates in the race are fellow shadow cabinet minister Yvette Cooper and Liz Kendall.
In an interview with Iranian Press TV show The Agenda, Mr Corbyn described the attack on the World Trade Centre, wars in Afghanistan and Iraq and the death of Bin Laden as tragedies, arguing "the solution has got to be law not war".
"There was no attempt whatsoever that I can see to arrest him and put him on trial, to go through that process," he said.
"This was an assassination attempt, and is yet another tragedy, upon a tragedy, upon tragedy."
Liberal Democrat leader Tim Farron said Mr Corbyn was "utterly wrong" for what he had said while Kevan Jones, Labour's defence spokesman, said: "This just shows you how out of touch he is with what most people's views are."
Conservative MP Nadhim Zahawi said: "Osama bin Laden was a terrorist who any sensible human being in the world would want either killed or arrested. For him to call this a tragedy and appear to compare it to what happened on 9/11 is frightening."
At-a-glance profiles of the four contenders
The interview is the latest in a series of past comments and associations the left-winger has been forced to defend since running for leader.
The majority of votes are thought to have already been cast, but fellow contender Mr Burnham told a rally in Pudsey, West Yorkshire, there were still tens of thousands of people yet to vote and it was still "all to play for".
"I will bring forward a bold manifesto that can win for Labour at the next general election, and that's the appeal that I'm making to all people undecided in this election."
He said his leadership would "unite the party" and Labour would no longer "dance to the tune of the Tory party".
The result of the leadership race will be announced on 12 September.

Thames Valley Police made the discovery after being called to the property in Iffley Road at 14:58 BST. They are treating the death as "unexplained".
The force said Iffley Road would be closed to drivers and pedestrians for a "number of hours" between The Plain and Circus Street.
Officers are appealing for information from the public.

Cooper, 18, was registered with the National League in time for him to be part of the squad for Saturday's home fixture against Braintree Town.
The son of former Swindon, Peterborough and Darlington manager Mark Cooper, the Blues under-21 player signed a first professional deal last summer.
However he is yet to make a senior appearance for Birmingham City.
Hyundai Motor will defer payments due from US federal employees affected by the partial government shutdown.
[NEXT_CONCEPT]
Jeremy Corbyn has been criticised for saying it was a "tragedy" that Osama Bin Laden was killed rather than being put on trial.
[NEXT_CONCEPT]
The body of a man has been found at a property in Oxford, police have confirmed.
[NEXT_CONCEPT]
Forest Green Rovers have signed midfielder Charlie Cooper from Birmingham City on a one-month loan.