The PM said the UK would need “an alternative way forward” after her plan was defeated by 58 votes on Friday.
MPs from all parties will test support for other options during a second round of “indicative votes” on Monday.
But government sources have not ruled out a run-off between whichever proves most popular and the PM’s Brexit plan.
Labour leader Jeremy Corbyn has called on Mrs May to change her deal or resign immediately, while Northern Ireland’s DUP – which has propped up Mrs May’s minority government – also continues to oppose the deal.
The government has so far failed to win over 34 Conservative rebels, including both Remainers and Tory Brexiteers who say the deal still leaves the UK too closely aligned to Europe.
However, a No 10 source indicated the prime minister would continue to seek support in the Commons.
They insisted efforts were “going in the right direction”, given the margin of defeat was down from 149 a fortnight ago.
Leave voters registered their anger at the latest rejection, on the day the UK was originally scheduled to leave the EU.
Thousands gathered outside Parliament to protest against the delay, bringing traffic to a standstill.
And the Conservative former Attorney General Dominic Grieve, who has campaigned for a further referendum on the deal, lost a vote of no-confidence in his Beaconsfield constituency.
There is every chance that the prime minister will again – with routes outside the normal boundaries – try to make a version of her Brexit deal the end result of all of this.
Despite a third defeat, despite the embarrassment of repeated losses, don’t imagine that she is ready to say a permanent farewell to the compromise deal she brokered with the EU or, straightaway, to her time in office.
There is still a belief in the heart of government that there could be a way round, perhaps to include the prime minister’s agreed treaty as one of the options that is subject to a series of votes that will be put in front of the Commons next week.
Mrs May has until 12 April to seek a longer extension to the negotiation process to avoid the UK leaving without a deal, which most MPs believe could harm business and create disruption at ports.
However, she said any further delay to Brexit was “almost certain” to involve staging elections to the European Parliament in May.
Downing Street later said this was not an “inevitability” but Justice Minister Rory Stewart told BBC Newsnight Friday’s vote had been “the last chance” to avoid that.
He said it would take a “miracle” – and the support of up to 150 Conservatives – on Monday for a majority of MPs to back a Brexit option that supported staying in the customs union.
This allows businesses to move goods around the bloc without checks or charges but continued membership would bar the UK from striking independent trade deals.
And BBC political correspondent Chris Mason said: “Leaving it was a Conservative manifesto commitment, and an about-turn on that could tear apart the party from the cabinet down.”
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