David Miliband has urged the UK to seek a "safe harbour" after Brexit by staying in the European Economic Area.
The ex-Labour foreign secretary said Jeremy Corbyn, who has ruled out the so-called Norway model, risked becoming the "midwife of a hard Brexit".
Ministers say EEA membership would require the UK to accept most EU rules as well as freedom of movement.
But Mr Milband told the BBC that the UK must get real, saying that 60% of UK trade was "under European aegis".
Mr Miliband, who has worked for the International Rescue Committee in New York since 2013, is joining politicians from other parties who favour retaining the closest links with the EU at a rally later.
He told BBC Radio 4's Today programme that time was running out for the UK to reach a deal which protected British economic interests and prevented the risk of "conflict" in Northern Ireland.
At a meeting later, Mr Miliband, former Lib Dem leader Nick Clegg and former Tory education secretary Nicky Morgan will urge Parliament to force the government's hand by voting for the UK to remain in some form of customs union as well as retaining full access to the single market after it leaves on 29 March 2019.
Mr Miliband said the move was about cross-party working in the national interest, not creating a new political movement or new centrist political party.
Many Labour MPs, and some Tories, favour the so-called "Norway model" of remaining in the EEA - which is an economic grouping of all EU countries as well as Norway, Liechtenstein and Iceland.
EEA membership would see the UK retain full access to the EU's internal market of 300 million consumers in return for making financial contributions and accepting most EU laws.
Free movement laws would also apply - so EU citizens could move to all EEA countries to work and live.
The government says this would be counter to the EU referendum, which Mrs May has described as "a vote to take control of our borders, laws and money".
Mr Miliband said EEA membership would allow the UK to have "structured" trade relations with the EU in goods and services, adding there was support from Norway and others for the UK to be a member.
While the EU accounted for 40% of the UK's direct trade with the EU, Mr Miliband said that if you took into account third-party EU trade agreements with other countries, the figure was about 60%.
"Membership of the EEA, as what I would call a safe harbour for Britain after Brexit, is on the table for every MP and party leader," he said.
Mr Miliband joined a growing chorus of senior Labour figures - including former Lord Kinnock - calling on Mr Corbyn to rethink his position ruling out the EEA option.
"If Jeremy Corbyn is not careful, he will be the midwife of a hard Brexit that will threaten the living standards of the very people that he says he wants to stand up to represent," he said.
He also suggested the UK had to take into account the "fundamental change" of direction that the US was heading in under President Donald Trump when it considered its future.
'Customs impasse'
The BBC's assistant political editor Norman Smith said Mr Miliband and others were calling on Parliament to take back control of the Brexit process amid cabinet divisions over future trade relations with the EU.
The prime minister is holding a series of meetings with Tory MPs in Downing Street later to set out the two customs options proposed by the government.
On Sunday, Environment Secretary Michael Gove said a customs partnership with the EU - believed to be the option favoured by Mrs May - had "flaws" and expressed doubts whether it could be delivered on time.
Foreign Secretary Boris Johnson has publicly described the model - which would see the UK collect tariffs on goods entering the country bound for the EU - as crazy.
The CBI employers group, which backs remaining in a full customs union with the EU, warned that the issue needed to be resolved "within days".
"If we don't break the impasse on this customs decision, everybody will be affected - manufacturers, services companies, retailers," said its director Carolyn Fairbairn. "An awful lot hangs on this now."