Rory Stewart won 37 votes in Tuesday's second-round ballot of 313 Conservative lawmakers

Rory Stewart has torn up the political rule book and used a quirky social media campaign to move yet another step closer Tuesday to becoming Britain's unlikeliest prime minister.

Few outside parliament had heard of the 46-year-old son of a former top foreign intelligence officer until he was named international development minister in May.

Fewer still gave him a serious shot at winning when he pronounced himself ready to succeed Theresa May as Conservative party chief.

But he managed to channel charm and a bashful charisma into a Twitter campaign that caught fire and made him the against-all-odds story of the leadership campaign.

Phone videos of him chatting to strangers about the complexities of Brexit were both amusing and fresh -- and have won him unexpected levels of support.

Stewart won 37 votes in Tuesday's second-round ballot of 313 Conservative lawmakers. That nearly doubled the 19 he earned in last week's first round vote.

He is now vying with three others for a chance to get into a likely runoff next month against runaway favourite Boris Johnson -- the former foreign minister and onetime London mayor.

"We seem to have almost doubled our vote again...more to come...#walkon," Stewart tweeted moments after the ballot.

'Lawrence Of Arabia'

Stewart combines a peculiar mix of big city savvy with an unabashed nostalgia for British traditions and a thirst for far-flung adventure.

He voted for Britain to remain in the European Union in the 2016 referendum and now wants to deliver a "pragmatic Brexit" that avoids a chaotic no-deal divorce.

He admits his message does not exactly chime with many hardline Brexiteers in the Conservative party.

But he counters: "I've got nothing to lose."

Stewart's proposed solution to the political deadlock -- a "citizens' assembly" in which a group of random Britons thrash out a compromise that can get parliamentary approval -- is redefining outside-the-box thinking.

"Rory Stewart's nerdiness means he is just the sort of oddball the British love," The Guardian newspaper wrote.

He "gives impassioned speeches in (parliament) about hedgehogs and sees himself as the second coming of TE Lawrence. What's not to like?"

"The Lawrence of Arabia" moniker derives from his career as a foreign officer and a book he wrote about his walk across Afghanistan "with only a toothless mastiff for company" in 2002.

He was appointed a deputy governor in southern Iraq after the US-led 2003 invasion and had previously served as Britain's representative in Montenegro after the 1999 Kosovo war.

Stewart has also worked in the UK embassy in Indonesia and run a charity in Afghanistan.

All this has led to speculation that Stewart -- like his father -- was a member of the MI6 foreign intelligence service. He denies it.

Tutor To Princes

Yet Stewart is anything but anti-establishment.

He attended both Eton and Oxford -- the blue-blood path taken by a large chunk of Britain's political and financial elite.

He told The New Yorker magazine in 2010 how Prince Charles once asked him to tutor his sons William and Harry when they were still schoolchildren.

Stewart said he spent "a lot of time" with Prince Charles during his Oxford years.

"That's really where my friendship with him began," he said of Britain's future king.

Stewart's career in politics began with his 2010 election to parliament. He has also served as prisons minister and overseen the cabinet's Africa brief.

"I am a Conservative. I'm very proud of being a Conservative. I believe deeply in limited government, individual rights, respect for tradition and love for my country," he said.

"I am the only candidate in this race who served my country in Iraq and the Balkans."

And Johnson "doesn't, as far as I can see, have a plan".