President Trump signed a new executive order Monday that temporarily bans visitors from six Middle East and African counties while the administration creates an extreme vetting program, taking pains to remove the pitfalls of the original order and the perception that it’s a “Muslim ban.”

The new order also dropped Iraq from the list of seven predominately Muslim countries covered by the original Jan. 27 executive order that was blocked by federal courts. The ban will be phased in over a 10-day period to avoid the chaos at airports following the original executive order that caught travelers in transit, and the wording has been changed to avoid suspicions that it is a “Muslim ban,” said a Department of Homeland Security (DHS) official.

“We are going to have a very smooth implementation period,” said the official, who spoke to reporters on condition of anonymity. “The motivation behind it is a real concern by the American people for their security.”

The 90-day halt of visitors from Iran, Libya, Somalia, Sudan, Syria and Yemen will take full effect March 16, according to the official.

The official stressed that the Trump administration was confident that the original executive order would have eventually prevailed in court, but for expediency the president opted for a new order that addressed the courts’ concerns. The inclusion of Iraq in the original ban raised objections from some U.S. military officials and the Iraq government because of special consideration previously given to Iraqis who aided in the war effort, including translators. The administration said Iraq was removed from the list, despite the presence of Islamic State fighters in the county, because of extra steps the Iraq government was taking to share information and assist the U.S. in vetting travelers.

“As a result of this increased information sharing, Iraqi citizens are not affected by the Executive Order. Of course, all normal immigration processing requirements continue to apply, including the grounds of inadmissibility that may be applicable,” according to the fact sheet.

In an attempt to demonstrate that the order is not a “Muslim ban,” which was a chief criticism of the original order based on Mr. Trump’s comments during the campaign, the new order removes language that exempted religious minorities from the targeting predominately Muslim countries.

The new order exempts people who currently have visas or legal “green card” status in the U.S., a carve-out that was not included in the original order.

The modifications did not satisfy Mr. Trump’s critics.

“This isn’t about keeping America safe. Let’s call it what it is. This ban is a racist and anti-Islamic attempt to divide us up,” tweeted Sen. Bernard Sanders, the Vermont independent who emerged as a leading voice for the left after his 2016 run for the Democratic presidential nomination.

Another court challenge is likely for Mr. Trump’s order. Omar Jadwat, director of the American Civil Liberties Union’s Immigrants’ Rights Project, said the new order had the “same fatal flaws” as the last one.

“The only way to actually fix the Muslim ban is not to have a Muslim ban,” he said. “Instead, President Trump has recommitted himself to religious discrimination, and he can expect continued disapproval from both the courts and the people.”

Jon Rainwater, executive director of the liberal activist group Peach Action, said the order was “the same Muslim ban with a fresh haircut, and a botched one at that.”

“It still blocks refugees and travelers from six Muslim-majority countries. It still violates core American values of religious tolerance and providing refuge to those fleeing violence and persecution. It still makes Americans less safe by feeding into the recruitment narratives of violent extremists,” he said. “There’s no evidence that banning travelers based on nationality actually protects anyone.”

The administration has delayed the signing as it works with federal agencies that will be responsible for enforcing the new order. The order on tougher vetting of travelers from terror-prone nations was one of Mr. Trump’s campaign pledges, and he signed the original order shortly after taking office. But a federal judge in Seattle blocked the action, and a panel of the 9th Circuit Court of Appeals upheld the injunction. Since then, the administration has been working on a new directive while defending the initial order in court. In an address to Congress last week, Mr. Trump said his actions are justified due to national security concerns.

The extreme vetting program will include:

• Uniform screening standards for all immigration programs government-wide, as developed by DHS in conjunction with the Director of National Intelligence and State and Justice departments.

• A biometric entry-exit system for all targeted travelers entering and departing the U.S.

• A review by the State Department of all non-immigrant visa reciprocity agreements to ensure that they are, with respect to each visa classification, truly reciprocal.

• New restriction on the Visa Interview Waiver Program and additional non-immigrant visa applicants will have to undergo in-person interview.

The executive order also temporarily halts the acceptance of new refugees into the U.S. for 120 days while DHS and other agencies review screening procedures to ensure refugees admitted in the future do not pose a security risk, according to a fact sheet provided by the Trump administration. In another change from the original order, Syrian refugees will not be subject to a permanent ban.

Upon resumption of the Refugee Admissions Program, the executive order calls for refugee admissions not exceed 50,000 for fiscal 2017, cutting by more than half the 110,000 refugees projected to be admitted this year under Obama administration guidelines.

Supporting claims that the U.S. is at risk because of its refugee policies, administration officials said that of the FBI’s 1,000 ongoing anti-terrorism investigations, 300 cases involved suspected terrorists who came to the country as refugees.

Unlike the public signing ceremony for the original order that Mr. Trump attended at the Pentagon on Jan. 27, the president signed the new immigration order at the White House without press coverage.