One group of Washington operatives not sorry to see Defense Secretary James Mattis leave is the Trump supporters who feel locked out of Pentagon jobs.

Insiders say Mr. Mattis’s chief of staff, retired Navy Rear Adm. Kevin M. Sweeney, preached a “non-political” Defense Department.

They translated his edict as meaning that Trump campaign people and Republicans on Capitol Hill would have limited access to 160 plum, high-paying appointments.

Mr. Mattis gave Mr. Sweeney wide authority over personnel matters, sources said. Mr. Sweeney’s official Pentagon biography states: “He is responsible for providing counsel and advice to the Secretary on all matters concerning the Department.”

Said a former Trump campaign adviser: “General Mattis was briefing Hillary during the 2016 campaign and only became a Trump supporter after the election. He then worked hard to keep Trump campaign advisers and surrogates out of the Pentagon while undermining Trump policies all along. I hope he doesn’t let the door hit him on the way out.”

Sources point to Michele Flournoy as the best example of Mr. Mattis‘ “non-political” department. She served as President Barack Obama’s undersecretary of defense for policy and is a Hillary Clinton supporter. Yet she was Mr. Mattis‘ first choice for the ultra-plum job of deputy defense secretary, to the exclusion of rank-and-file Republicans who had been outside looking in for eight years.

Perhaps the Flournoy near-hiring contributed to Mr. Trump calling Mr. Mattis “some sort of a Democrat” on CBS’ “60 Minutes.”

After the Mattis announcement last week, Ms. Flournoy went on NPR to say, “We should be very concerned. Jim Mattis brought a lot of experience and perspective to the job and to a president who has been known for his unpredictability and his impulsiveness and his rash decisions.”

The deputy secretary job ultimately went to Patrick M. Shanahan, a longtime Boeing executive outside the Republican foreign policy sphere. He now will lead the Pentagon as acting secretary beginning Jan. 1 until a new chief is confirmed by the Senate.

“Mattis mostly hired Democrats, so this is an opportunity for a Trump do-over of political appointees at the Pentagon,” tweeted Christian Whiton, a former senior adviser at the State Department. “Would be nice if it happens.”

One knowledgeable source said that the way to understand Mr. Mattis‘ personnel decisions is that he prefers people he knows or has worked with, as opposed to strangers from a conservative think tank.

This source said he approved James F. Geurts as assistant Navy secretary for acquisition because he knew of Mr. Geurts’ weapons-buying work at U.S. military commands.

Mr. Sweeney, a former carrier strike group commander, worked for the defense secretary as a close aide at U.S. Forces Command and then at U.S. Central Command before Mr. Mattis plucked him out of the private technology world to become his chief of staff.

A second source defended Mr. Sweeney. While he stresses a non-political Pentagon, Mr. Sweeney has asked for Republican referrals, only to run up against a tide of “Never Trumpers” who would be dead-on-arrival with White House gatekeepers, this insider said.

“Sweeney has leaned on people to give him names,” said the source, adding that any anti-Trump tweet or blog might disqualify a candidate.

As an example of Republican hiring, this source pointed to Robert Hood, assistant secretary of defense for legislative affairs. He learned about Washington as an aide to House Speaker Newt Gingrich and held several posts under President George W. Bush.

Much of the Washington GOP establishment went to war against candidate Trump and supported his party rivals such as former Florida Gov. Jeb Bush and Sen. Marco Rubio. In August 2016, for example, 50 Republican ex-national security figures signed a protest letter.

“He is unable or unwilling to separate truth from falsehood. He does not encourage conflicting views. He lacks self-control and acts impetuously. He cannot tolerate personal criticism. He has alarmed our closest allies with his erratic behavior,” they said. “All of these are dangerous qualities in an individual who aspires to be President and Commander-in-Chief, with command of the U.S. nuclear arsenal.”

The signers included George W. Bush’s CIA director, retired Air Force Gen. Michael Hayden, who has remained a sharp critic of Mr. Trump’s demeanor and foreign policy.

But not all Republicans shied away from Mr. Trump: Former Secretaries of State Henry Kissinger, a Trump adviser, and James A. Baker abstained.

The Washington Times asked chief Pentagon spokeswoman Dana W. White about complaints that conservative Trump supporters faced hurdles.

“I can personally attest to Kevin Sweeney’s tireless efforts to recruit qualified Republicans for positions within the Pentagon,” Ms. White said. “Unfortunately, much of the Republican national security establishment removed themselves from consideration as self-proclaimed ‘Never Trumpers.’ Therefore, it has been more challenging identifying and vetting politicals for key national security positions.”

Ms. White herself carries solid conservative credentials as a former press aide to House Republicans, an adviser to the 2008 John McCain presidential campaign and program director at the Heritage Foundation.

The former Trump campaign worker said many believe the president turned his back on scores of people who devoted their lives to him and have suffered through the Trump-Russia investigation.

“General Mattis is a cautionary tale for conservatives thinking about getting involved in 2020 re-election campaign,” the ex-adviser said. “Numerous advisers who took significant risks in supporting Trump during the 2016 campaign had their lives destroyed by a cast of thousands. Meanwhile, Mattis fought hard to keep them out of the Pentagon, adding insult to injury.”

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