Mortgage holidays and jobs furloughing are to end on Saturday, with other support measures starting.
But there are still gaps in support that need filling, some organisations have said.
The Treasury said it had put in place a number of generous support schemes for individuals and businesses.
The mortgage holiday scheme introduced at the start of the Covid-19 crisis ends on Saturday as does the job furlough scheme, which is being replaced by the Job Support Scheme.
It will leave a fifth of mortgage holders - around 1.6 million households - worried about paying their mortgage over the next three months, according to the Joseph Rowntree Foundation.
The poverty campaign charity said: "There is a real risk that mortgage-holders on low incomes will be pulled into poverty and hardship."
It said 890,000 working households with a mortgage expect to see a drop in earnings over the next month, but 85% of them - 750,000 households - aren't eligible for any government support with their housing costs.
"It's not right that during a time of huge uncertainty, many households are discovering that they are excluded from the only lifeline that could help meet their housing costs," said Darren Baxter, policy and partnerships manager at the charity.
The Joseph Rowntree Foundation wants the Support for Mortgage Interest payment to be reformed to help people who lose their jobs to keep their homes as they weather the coronavirus storm.
Self-employed homeowners could face a double hit as their grants are reduced, the Labour party said.
"Self-employed homeowners are facing a perfect storm because the government has decided to abandon them just as we head into the winter," said shadow chancellor Anneliese Dodds.
"There's still time for the government to stop a bleak winter for Britain's self-employed workers.
"It must remove the mortgage cliff edge, fix the gaps in its income support schemes, and help people defer the cost of interest payments."
From November, self-employed people will be able to apply for a third grant under the Self-Employment Income Support Scheme (SEISS) to cover the next three months, worth 40% of their pre-virus trading profits.
But that figure is down from the 80% offered during the first grant and 70% during the second application period.
People renting their home are more likely to have fallen behind with their housing costs than mortgagers, according to the Resolution Foundation, which campaigns on living standards.
Its research suggests nearly one-in-eight private renters and more than one-in-six social renters are currently unable to cover their housing costs in full.
"Renters are being particularly badly hit," said Lindsay Judge, research director at the Resolution Foundation.
"They are much more likely to have lost their jobs and significant numbers are only managing by cutting other expenditures, drawing down on savings or getting into debt in order to meet their rent."
She called on policymakers to ensure the social security system supports struggling families effectively over the coming months and take urgent action to avoid an increase in homelessness.
"But landlords might also need to recognise that as household incomes fall, rents are more likely to need to go down than up," she added.
Landlords, however, have called on the government to give the sector extra help to protect those who are renting.
The National Residential Landlords Association warned that 300,000 renters could be at risk of losing their jobs as the furlough scheme closes.
It said renters under the age of 35 will face the brunt of the crisis as those who find themselves on benefits will only be able to claim for a room in a shared house.
"With rates of Covid-19 rising, we need to do everything possible to sustain tenancies," said Ben Beadle, chief executive of the NRLA.
"The government needs urgently to step in and fund renters who are struggling as a result of the pandemic."
The Treasury said the new Job Support Scheme which starts on Sunday, combined with the Job Retention Bonus, will cover at least 95% of the total employment costs for average previously furloughed employee until February.
"The Job Support Scheme will continue to protect jobs throughout the difficult months ahead and is part of our comprehensive Plan for Jobs," said Chancellor of the Exchequer, Rishi Sunak.
The Treasury added: "The JSS and JRB are just one part of our generous package of measures, that includes the extended business grants and Self-Employed Income Support Schemes announced last week, which will continue to support businesses and livelihoods across the country over the winter months."
Rule No. 1: Never lose money; rule No. 2: Don’t forget rule No. 1.