(CNN) President Donald Trump will now wait until next week to unveil his new attempt to impose a temporary travel ban on citizens of states he says pose a high risk of terrorism.

An administration official said the order, which represents a do-over after Trump's first attempt was blocked by federal courts, would come by the middle of the week. The order had been expected to emerge sometime this week.

The order will form the second thrust of a new administration push to significantly overhaul the shape of the American immigration system, following the release of new memos Tuesday empowering state and local authorities to enforce laws that could eventually lead to mass deportations.

It also marks an important moment for Trump's vision of an expansive executive presidency as he contemplates other areas of sweeping policy action.

The significance of this new attempt -- the language of which is expected as soon as Wednesday -- is reflected in the participation of White House Counsel Don McGahn.

McGahn's office had only a cursory look at Trump's original order, which was written by transition and policy staff. Significantly, Trump's key political aide Stephen Miller has had much less to do with the second executive order, sources familiar with the matter said, and the Trump administration was communicating with Republicans on Capitol Hill about the legislation.

Another administration official disputed the notion that Miller is less involved now, saying he is in fact more involved in this process than the first one.

Trump's initial attempt to install a travel ban -- one of his fundamental campaign promises -- was a disaster, halting the administration's fast start in its tracks.

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The move temporarily blocking citizens of Syria, Sudan, Yemen, Iran, Iraq, Somalia and Libya from entering the US unleashed a weekend of chaos at the nation's airports, after the hurriedly drafted and poorly implemented order caused confusion among border and customs officials about what it actually meant and which classes of travelers were included.

His order was quickly halted by the federal courts in a first showdown between his strong-arm executive powers and the judiciary -- leading Trump to belittle judges on Twitter.

The showdown inflicted an early blow on the reputation of the new White House and claims that Trump's expertise as a master dealmaker and businessman would make up for his inexperience in Washington and governance as he set about fundamentally transforming America.

Stung by the scorn of the courts and the political world, the White House eventually retreated to plot a new approach -- one that is likely to be considerably narrower than the initial version. Permanent US residents, or green card holders, for instance, are expected to be exempted from the ban.

Trump rarely admits an error or apologizes for a misstep. But even he realizes that there is not much he can do but frame a new executive order to satisfy the 9th Circuit Court of Appeals' decision upholding a stay on the order imposed by a federal judge in Seattle.

His spokesman Sean Spicer said on Wednesday that the administration would tailor the new order to the ruling of the 9th Circuit and that the new plan was "basically completed."

"I think what we're now doing is working with the various agencies and departments to make sure that the implementation of that is done in an extremely smooth way," Spicer said.

It is vital for the credibility of the President and a White House that things do indeed go smoothly this time around.

This may be the best, last chance for the administration to establish whether it can write an executive order that can honor Trump's goals but at the same time not fall foul of constitutional due process rights of travelers trying to get into the United States who might be covered by the ban.

The White House faces a high bar in drafting the new order because its constitutional interpretation has already proven open to being challenged. The 9th Circuit, for instance, rejected the administration's argument that the judiciary lacked the authority to block the travel ban as "contrary to the fundamental structure of our constitutional democracy."

Apart from the legal minefield the new order must traverse, the Trump White House has huge political credibility tied up in the travel ban.

A repeat of angry demonstrations at airports, with tales of travelers being turned back after getting on planes with what they thought were valid visas, would deal another blow to the new administration.

Has White House learned a lesson?

By Tuesday, there were initial signs that the administration had started to conduct the widespread consultation among its allies in the House and the Senate that was missing with the initial travel ban.

A top House Republican aide said the Department of Homeland Security had been in touch with House Speaker Paul Ryan's office about the new executive order. But it is unclear whether the White House has been engaging with the committees with jurisdiction over immigration and national security.

Keeping allies in the loop is important because the refashioned executive order is likely to be viewed elsewhere in Washington as an omen for whether the administration is able to pull off the technical, legislative and political lifts needed to implement the promises for fundamental changes in America that Trump vowed as a candidate.

Of course, there is certain to be anger and controversy once the new order is released -- its opponents will brand it unconstitutional and un-American and the affected nations are likely to vocally protest and may take counter-measures. Lawsuits are already promised.

But a smooth implementation process could at least preserve the administration political space to press ahead with a scheme that does garner support from many Americans, especially Trump supporters.

It would also spare Capitol Hill Republicans from media interrogations about the White House's ham-handed unveiling of the original order.

There is intense interest in Washington and around the world about how the second executive order will differ from Trump's initial attempt.

Going forward, a new order could also only apply to people who are not yet in the visa process. The new order may apply to all seven nations originally named and could retain a ban on the entry of Syrian refugees, a source on Capitol Hill said.

This story has been updated to reflect the latest reporting.