"This order, properly construed, should really allow for only the narrowest implementation of any part of the ban," Omar Jadwat, director of the ACLU's immigrants' rights project, said during a conference call with reporters.
Refugees awaiting resettlement through American non-profits, current visa-holders, students seeking to attend American universities, foreigners hired to work on US soil, and the relatives of those already living here - among other examples - should be exempt from any enforcement of the ban, according to their reading of the Supreme Court's decision, the advocates said.
But other groups, like Amnesty International USA saw it differently, and warned that the court's decision would lead to a fresh bout of "chaos" at airports, and an enforcement of the ban that would "tear families apart".
And human rights group called on Congress "to step in immediately to nullify [the ban] once and for all"- or risk further chaos.
"Reinstating any part of this ban could create chaos in the nation's airports and tear families apart," Margaret Huang, the group's executive director, said in a statement.
The International Rescue Committee, which provides refugee assistance, said the court's decision will impact on "innocent, security-vetted refugees".
"The Court's decision threatens damage to vulnerable people waiting to come to the US: people with urgent medical conditions blocked, innocent people left adrift, all of whom have been extensively vetted," said the organisation's president and former British Foreign Secretary, David Miliband.
And the Council on American-Islamic Relations (CAIR), a Washington-based Muslim advocacy group, said the court's decision "ignores the Islamophobic origins of the policy and emboldens Islamophobes in the Trump Administration".
Trump's first iteration of the ban in January sought to block entry to refugees, immigrants and other nationals-including permanent legal residents of the US - who held citizenship in Yemen, Syria, Iran, Iraq, Sudan, Libya and Somalia. The ban's blanket enforcement set off a wave of detentions, deportations and the mass revocation of visas, and it provoked chaotic scenes of protest and confusion at airports across the country.
The ACLU and other advocacy groups involved in lawsuits against the ban said they didn't expect the same chaos to ensue this time because the Supreme Court's decision had restricted the policy so much that it would have an impact only a small number of people.
"It should only apply to pending visas. And in the entire universe of pending visas, there is only one that doesn't involve a tie" to someone or something in the US - a tourist visa, said Becca Heller, the director of the International Refugee Assistance Project (IRAP), the plaintiff in IRAP vs Trump, the full case of which the Supreme Court will hear in October.
But Heller and other advocates also said they will closely monitor the enforcement of the limited travel ban, given the possibility that the Administration may interpret the court's decision differently. They said refugee and immigrant advocacy groups would be prepared to file new lawsuits over any perceived abuses.
"It's going to be really important to us over this intervening period to make sure that the government abides by the terms of the order and is not going to use it as a back door to implement the full-scale Muslim ban that it's been seeking to implement throughout the presidency," said Jadwat.
"I think the Government will be shooting itself in the foot if it does decide to take this as some kind of license to apply the ban aggressively," he added.
