Former House of Commons leader Andrea Leadsom and former Brexit Secretary Dominic Raab joined the fray Saturday night. Both had earlier resigned from May's Cabinet to protest her Brexit policy.
Health Secretary Matt Hancock said Saturday morning he is seeking to replace May, joining several others who have announced they will run to become the Conservative party's next leader, and by default, Britain's new prime minister.
May announced Friday she plans to step down as Conservative Party leader on June 7 and remain as a caretaker prime minister while the party chooses a new leader in a contest that officially kicks off the following week.
She plans to remain as party leader through U.S. President Donald Trump's upcoming state visit and the 75th D-Day anniversary celebrations on June 6.
Her successor will have to try to complete Brexit - a task that May failed to deliver during her three years in office. While she succeeded in striking a divorce deal with the European Union, the plan was defeated three times in Parliament by British lawmakers from across the political spectrum.
The EU extended Britain's departure date to Oct. 31 but there still is no consensus among British lawmakers about how or even if the country should leave the bloc.
Even before a new leader is chosen, the Conservative Party is expected to fare poorly when the results of the European Parliament election in Britain are announced Sunday night.
The best-known contestant for the Conservative leadership post is former Foreign Secretary Boris Johnson, who has said he will take Britain out of the EU on Oct. 31 even if no deal has been reached with EU leaders.
Johnson's willingness to back a no-deal Brexit is already causing some ripples.
Another Conservative contender, International Development Secretary Rory Stewart, said Saturday that he could not serve in a Cabinet under Johnson if Johnson wins. Stewart says he could not work for a leader who is comfortable with the idea of a no-deal Brexit.
Stewart complained that Johnson said in a private meeting several weeks ago that he would not push for a no-deal departure but appears to have changed course completely.
Many economists and business leaders have warned that a no-deal departure would have a drastically negative impact on Britain's economy and also hurt its European neighbors.
The field is likely to grow to about a dozen candidates, with a winner expected to be chosen by mid or late July. Senior Conservatives including Home Secretary Sajid Javid and Foreign Secretary Jeremy Hunt are among those considering a leadership run.
The Conservative Party chooses its leaders in a two-step process. First there's a series of votes among the party's legislators to establish two top contenders, then those names are submitted to a nationwide vote by about 120,000 party members.
The winner becomes party leader and prime minister, although the opposition Labour Party is warning of an immediate challenge to the new leader with an eye toward forcing an early general election.
John McDonnell, Labour's economic spokesman, told the BBC on Saturday the party would push a no-confidence vote against the new prime minister right away.
"We believe any incoming prime minister in these circumstances should go to the country anyway and seek a mandate," McDonnell said.
An earlier Labour Party attempt to force an early election failed in January when May's government survived a no-confidence vote.
The U.K.'s next general election will be held in 2022 unless a government collapse speeds up the timetable.
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