White House chief of staff John Kelly has already begun to impose order and structure on President Trump's occasionally chaotic team, but his authority has not yet extended to one of the most unruly aspects of Trump's presidency: @realDonaldTrump.

Trump gave Kelly unchecked hiring and firing power over White House aides below the senior counselor level when he chose the former homeland security secretary to replace outgoing chief of staff Reince Priebus late last month. Only a handful of top advisers — including Jared Kushner, Ivanka Trump, Kellyanne Conway, and chief strategist Steve Bannon — were exempted from Kelly's reach by requiring Trump's permission to remove or reassign, a source close to the White House said.

But while the president has offered Kelly a level of control Priebus never managed to obtain, Trump has resisted giving his new chief of staff veto power over the spontaneous and provocative tweets that often serve as a distraction for his administration.

A series of news reports suggesting Kelly had sought oversight of Trump's Twitter account, including a report that claimed Kelly wanted to know in advance what the president planned to post, made their way to Trump's desk last week, a person familiar with the situation told the Washington Examiner.

Trump "was pissed when he read Kelly wanted to control his Twitter feed," the person said.

Kelly has had luck in other areas. Trump's "Presidential News Summary," a collection of press clips presented to him by his staff, "is even more tightly controlled" under Kelly, the person noted. Beyond what he views on cable news, which is more difficult for his staff to police, Trump sees even less of the alternative right-wing news outlets than he did when Priebus and outgoing press secretary Sean Spicer oversaw the Presidential News Summary, the source said.

And Kelly has cracked down on who can share information with the president, even regulating Trump's contact with his own Cabinet secretaries.

"While sending a memo to the president is now largely impossible, even print-outs of news stories are not permissible," the source explained. "Cabinet members have been told not to call or send any memos to the president and that they must now communicate only through Kelly."

But while Kelly controls more of the inputs that reach Trump, he is less in control of Trump's Twitter output. Trump flexed his muscles Monday with a flurry of tweets criticizing major news organizations, blasting "fake news" stories and taunting Democratic Sen. Richard Blumenthal of Connecticut by inviting him to "take a nice long vacation in Vietnam."

The Twitter rampage stoked speculation that Kelly has not yet consolidated control of Trump's social media habits, even as he gains greater authority over other aspects of the White House's operation.

Kelly did approve national security adviser H.R. McMaster's request last week to fire an intelligence aide with close ties to Bannon, a move that ruffled feathers among Trump's populist supporters. A senior administration official told the Washington Examiner that the termination had nothing to do with the aide's political allegiances and instead resulted from "inappropriate" behavior.

The decision exposed an ideological rift in the West Wing between Bannon and McMaster that several sources said had simmered for months before Kelly's arrival. At least in these areas, people see Kelly making improvements.

Priebus suffered criticism from inside and outside the White House over his inability to control who had contact with Trump and his failure to police internal feuds over influence and access. Although the former Republican National Committee chairman found himself at the center of speculation that he would soon leave his chief of staff position since the early days of Trump's presidency, Priebus remained at the top of the White House staff hierarchy for six months.

Now, White House aides hope that Kelly will instill a sense of order in the West Wing after a period of upheaval that saw several high-profile departures, including Spicer, former White House communications director Mike Dubke and his short-lived successor, Anthony Scaramucci.

"The internal White House dynamics are becoming clearer," said one person close to the administration. "People aren't distracted with Priebus' ineptitude anymore."