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Even as President Trump faces blistering criticism from Democrats for his executive order restricting immigration into the U.S., he is also encountering significant pushback from immigration hawks -- who argue the order falls short of the moratorium they were expecting, and are demanding Trump “do better.”

“The fuss turns out to have been unwarranted. The measure on its own will have little effect,” Center for Immigration Studies’ (CIS') Executive Director Mark Krikorian wrote in an op-ed that called the order “barely a start.”

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Trump signed the order Wednesday after promising Monday that he would “temporarily suspend immigration into the United States.”

"In order to protect our great American workers, I’ve just signed an executive order temporarily suspending immigration into the United States,” the president said during the coronavirus task force briefing at the White House. “This will ensure that unemployed Americans of all backgrounds will be first in line for jobs as our economy reopens."

The order cites “the impact of foreign workers on the United States labor market, particularly in an environment of high domestic unemployment and depressed demand for labor” as a reason for the restriction -- as well as pressures on health care and other factors amid the coronavirus crisis.

To hawks, it had initially appeared to be the immigration moratorium on all employment-based immigration they had been urging for weeks. Former Attorney General Jeff Sessions had called for such a moratorium of all employment-based immigration days earlier.

But the text of the order, which expires in 60 days, specifically applies to those seeking green cards from outside the U.S. -- including those applying under the diversity lottery, work green cards and chain migration.

It carves out exceptions for those coming in as health care professionals or for other coronavirus-related work, those applying under the EB-5 Immigrant Investor Program, spouses or children of U.S. citizens, members of Armed Forces and any immigrant deemed to be “in the national interest.”

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Most of the order is theoretical at the present moment. The State Department last month suspended all regular visa services for both immigrant and nonimmigrant visas -- with exceptions for agricultural worker visas -- so few visas are being processed anyway. But that suspension is expected to expire before Trump’s order.

It also comes on top of other immigration-related measures, such as restricting travel at land borders, slapping travel bans on China, Iran and the E.U., and turning away asylum-seekers and illegal immigrants at the border.

As with almost all of Trump’s moves to restrict immigration, it sparked fury among Democrats and pro-immigrant activists who accused Trump of fanning the flames of anti-immigrant sentiment and of causing havoc and fear for those who were on track for their green card.

"Make no mistake: this executive order is not about protecting American workers. The only thing it really accomplishes is keeping families apart,” House Judiciary Committee Chairman Jerrold Nadler, D-N.Y., and Immigration and Citizenship Chairwoman Zoe Lofgren, D-Calif., said in a statement. “It is just an excuse to advance President Trump’s and [White House adviser] Stephen Miller’s anti-immigrant agenda.”

But from those consistently calling for less immigration into the country, the criticism was also fierce. The Center for Immigration Studies’ Jessica Vaughan calculated that the pause would only apply to about 5 percent of annual admissions.

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Most problematic for those groups is the decision not to include restrictions on temporary and guest worker visas -- including the controversial H-1B visa program, which brings in tech workers.

“The health emergency will end when people stop dying from the virus. The economic crisis will likely last far longer,” Dan Stein, president of the Federation for American Immigration Reform (FAIR), said in a statement. “The president’s decision to put a temporary halt on immigration was a prudent act in light of the unprecedented job losses. But it will be rendered meaningless if large flows of guest workers continue unabated.”

Roy Beck, at NumbersUSA, argued that foreign guest workers are not technically immigrants, therefore Trump “technically is not backtracking on his promise to pause immigration when he allows guest worker visas to continue.” But he accused others in the White House of diluting the order from its original form.

“Unfortunately, the corporate lobbyists and other immigration expansionists in the White House persuaded the President to significantly water down the Proclamation that was supposed to carry out those lofty spoken principles,” he said, after citing tougher remarks by the president.

Others were not so sparing.

“Head. Bang. Desk. This is a cynical betrayal of the burgeoning ‘America First’ movement,” conservative commentator Michelle Malkin wrote. “It’s all moratorium hat and no cattle.”

Malkin also had no doubt as to whom she ultimately blamed: “Trump signed it. It’s on him.”

Now, restrictionists are eyeing a section in the order that says that the secretaries of State, Labor and Homeland Security will in 30 days “review nonimmigrant programs and [recommend to Trump] other measures appropriate to stimulate the United States economy and ensure the prioritization, hiring and employment of United States workers.”

They believe this could be the opportunity to add a ban on certain guest worker programs like the H-1B visa program, and are letting the president know.

“Failing to pause this program before the next cohort of foreign workers arrives October 1 would make a mockery of the president's claim to want to preserve American jobs for American workers,” CIS' Krikorian said.

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In a letter to Trump to be sent later Thursday, FAIR’s Stein urged Trump to use the 30-day review period to strengthen the order.

"These guestworker programs have spun completely out of control," Stein wrote. "This represents a deadly dependency that's wreaking havoc on the normal process by which our American workers are recruited into various components of the labor force.”

“President Trump, you promised you would do better, and as the EO states, you have 30 days to do so," he added.