Dominic Raab, who tried to vie with Johnson for the votes of committed Brexit supporters, got 30 votes, three short of the threshold needed to go through to the next round.
May stepped down as Conservative leader earlier this month after failing to secure Parliament's approval for her Brexit deal.
He also caused controversy by refusing to rule out suspending Parliament to thwart attempts by MPs to block a no-deal exit.
There were 10 candidates at the start of the leadership contest.
Mr Hunt again came second in the latest ballot of 313 Tory MPs, with 46 votes - three more than in last week's first vote.
International Development Secretary Rory Stewart, who began the contest as a rank outsider but built momentum with calls for realism and compromise on Brexit, said he was talking to Environment Secretary Michael Gove about combining forces.
Mr Gove has said that he would plan to get a new Brexit deal from Brussels that he would then bring back to Parliament, something Mr Stewart has said would not be possible. "He has star quality you don't often see in politics".
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The BBC's political editor Laura Kuenssberg said there was all kinds of possibilities regarding potential alliances and pacts between the candidates chasing Mr Johnson.
Hunt, who got 54 votes, and Gove with 51 are tussling for second place and the slot of Johnsons' runoff opponent.
She told the BBC the foreign secretary remained "a serious candidate for a serious job".
As the candidate who now has the fewest number of backers, there are rumours Mr Javid could drop out, but supporter Stephen Crabb told the BBC his favoured candidate had performed well in the debates and was not about to quit.
But Foreign Secretary Mr Hunt said people accused the Tories of being "the party of the rich" and "we must never fall into the trap of doing tax cuts for the rich and confirming that prejudice".
After Conservative MPs whittle them down to two candidates by the end of Thursday, the final pair will be put to a ballot of 160,000 party members, with the victor taking over in Downing Street likely in late July.
Two rounds of voting are scheduled to whittle the four down to two, though Johnson, barring a spectacular collapse in support, already has more than enough backing to ensure he cruises through.
The mainly pro-Brexit Conservative Party members, some 160,000 in all, will cast the deciding votes in a weeks-long process involving mail ballots that will culminate in the announcement of the victor the week beginning July 22.