May stressed that Britain can still leave the EU without taking part in the European Parliament elections if lawmakers approve her deal by May 22.
Asked about the implications of a Tory leadership contest, Mr Hammond insisted Theresa May will stay Prime Minister until she has taken Britain out of the EU.
The further delay comes after an initial extension pushed Brexit back from March 29 to April 12.
Sir Bill Cash, one of parliament's most long-standing anti-EU MPs, urged the PM to quit as she updated the Commons this afternoon on Wednesday evening's summit in Brussels.
May's own authority has been gravely compromised by the long Brexit ordeal and she has promised to step down once Britain leaves the bloc - if efforts to get rid of her more quickly do not bear fruit. The former U.K. Independence Party leader said delays to Brexit were "a willful betrayal of the greatest democratic exercise in the history of this nation". Britain has the option of leaving before that if Mrs May succeeds in getting a withdrawal agreement ratified by Parliament.
The BBC's Katya Adler said that, if the 31 October extension is confirmed, French President Emmanuel Macron "won the day" as his was the most hardline voice in the room.
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Under a deal struck during late night talks in Brussels, Britain received a "flexible extension" until October 31, saving all sides from a chaotic no-deal British departure at the end of the week.
"There may be some candidates who might be deterred from an application because of the political debate around Brexit, which inevitably the governor of the Bank of England can't avoid being part of", he said. No bulletin was issued after the talks, but in a brief statement McDonnell said they had been positive and constructive.
"It is no secret that the Labour leader has ducked and dived on the issue of a second European Union referendum, but at this crucial crossroads it is high time Jeremy Corbyn commits to putting a second referendum on the table as the only way forward".
Labour, which seeks to retain close economic ties with the European Union after Brexit, accuses the government of failing to offer concrete changes to its Brexit blueprint.
The dragging out of Brexit comes after May declared that "Brexit means Brexit, and we're going to make a success of it" July 11, 2016, when she was leader of the Conservative Party.
"I think we have reached a position where we are aware we are a million miles away from the promises made during the referendum campaign, and in those circumstances what the British public should be given is a final say".