LONDON (Reuters) - Britain’s main opposition Labour Party backs “the reasonable management of migration”, its leader said on Tuesday in a speech aimed at winning back millions of traditional supporters who backed Britain’s split from the European Union.

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But Jeremy Corbyn fudged a planned commitment that his pro-EU party was no longer “wedded” to the idea of free movement - a pillar of the European Union that, he said, could not be ruled out if it helped Britain win trade access.

In his first major speech this year, Corbyn set out his Brexit strategy more than six months after Britain voted to leave the EU to try to silence critics who say he has failed to challenge Prime Minister Theresa May with an alternative plan.

By offering workers higher wages and protection against undercutting by ruthless bosses, Corbyn hopes to ease concerns among Labour voters who feel the party is not in tune with their fears over immigration and to staunch their exodus to parties such as the anti-EU UK Independence Party.

“Labour is not wedded to freedom of movement for EU citizens as a point of principle, but I don’t want that to be misinterpreted, nor do we rule it out,” he told supporters in the English city of Peterborough, which voted strongly in favor of leaving the EU.

“We support fair rules and the reasonable management of migration as part of the post-Brexit relationship with the EU, while putting jobs and living standards first in the negotiations.”

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Since becoming party leader over a year ago, Corbyn has struggled to put his stamp on a party which has been pro-EU for more than two decades and was blindsided on Brexit by the more organized ruling Conservative Party, which enjoys a slim majority in parliament and has largely driven the debate.

A socialist on the left of his party, Corbyn has criticized the bloc for being in thrall to big business and said he would find ways to close the income gap between bosses and their workers by addressing pay ratios.

Corbyn had said earlier he was in favor of a cap on executive pay, but questioned after his speech he said that idea was one of a number alternatives.

He said looking at pay ratios would “encourage wage rises lower down and ensure a better sharing of resources”.

Supporters hoped Corbyn’s speech would “rebrand” the veteran campaigner as a more populist figure to boost ratings that have been dwarfed by those of May’s Conservatives. The latest poll gives the Conservatives a 14 point lead with 42 percent.

But despite acknowledging that many voters supported Brexit because of high rates of immigration, Corbyn would not be drawn on whether he wanted levels of immigration to fall.

Labour has long backed freedom of movement - one of the EU’s main principles - which the bloc says must be respected if Britain is to maintain its preferential access to the single market of 500 million consumers.

Corbyn said migration levels should depend on what kind of preferential access Britain would get to the single market.

“We cannot afford to lose full access to the European markets on which so many British businesses and jobs depend,” he said.