In a direct appeal to Chancellor Rishi Sunak, TUC General Secretary Frances O'Grady said the pandemic would not end in October, so neither should government support for jobs.
She pledged to work with Mr Sunak, as unions did in setting up the furlough scheme, to stop "the catastrophe of mass unemployment" and urged him: "Don't walk away."
Ms O'Grady's plea came in her keynote speech at a two-day COVID-secure TUC conference in London, with a small invited group in Congress House and union members across Britain joining online.
The Labour leader Sir Keir Starmer was due to speak in person and take questions from union members on day two of the conference, but that is now in doubt after he announced he was self-isolating.
The TUC conference began with a debate on the impact of coronavirus, with union leaders complaining of abuse against shop workers and claims of employees being forced back to work in unsafe conditions.
Ridiculing the government's handling of the pandemic, Ms O'Grady declared: "A useless app, a mutant algorithm and a half-baked test and trace system. Less 'moonshot', more a case of moonshine."
Ms O'Grady also hit out in her speech at claims that the chancellor is poised to scrap the rise in the minimum wage from £8.72 to £9.21 due next April because he cannot afford it after coronavirus.
And the TUC chief began her speech by fiercely criticising the GMB union after an inquiry found it was "institutionally sexist", warning: "This is a #MeToo moment for our unions."
In her plea for a furlough extension, Ms O'Grady said: "If the government doesn't act we face a tsunami of job losses.
"The chancellor must learn the lessons of the job retention scheme and keep on supporting jobs.
"So my message to the chancellor is this: We worked together once before.
"We are ready to work with you again - if you are serious about stopping the catastrophe of mass unemployment.
"Rishi Sunak: stand by working families - don't walk away."
Responding to the TUC leader's call, a government spokesman said: "Supporting jobs is an absolute priority which is why we've set out a comprehensive Plan for Jobs to protect, create and support jobs across the UK by providing significant, targeted support where it is needed the most.
"We are continuing to support livelihoods and incomes through our £2 billion Kickstart scheme, creating incentives for training and apprenticeships, a £1,000 retention bonus for businesses that can bring furloughed employees back to work, and doubling the number of frontline work coaches to help people find work.
"We are also supporting and protecting jobs in the tourism and hospitality sectors through our VAT cut and last month's Eat Out to Help Out scheme."
But Ms O'Grady said: "The pandemic isn't scheduled to end in October so neither should state support for jobs.
"When the crisis began, the chancellor said he would do 'whatever it takes'. He must keep that promise.
"Some will ask can the country afford to do it? The answer is - we can't afford not to."
Demanding the national living wage rise goes ahead as planned, Ms O'Grady said: "Coronavirus is no leveller. It has exposed huge inequality in modern Britain.
"The minimum wage - the wage of two million key workers - must rise as planned. Don't punch down on the working poor."