According to the new order, people without “close” family or business relationships in the US could be denied visas and barred entry.
The rules apply to people from Iran, Libya, Syria, Somalia, Sudan and Yemen, as well as to all refugees.
Lawyers took up positions as US airports, offering free advice.
However, there was no sign of the chaos that affected travelers when the first version of the ban was brought in at a few hours’ notice in January.
That is because the executive order does not affect people who already have valid visas or green cards.
The Department of Homeland Security said it expected “business as usual at our ports of entry”.
The Supreme Court ruling on June 26 upheld the temporary ban, a key Trump policy.
However, the judges provided a major exception, for those who have “a credible claim of a bona fide relationship” with someone in the US.
The effect is that citizens of the affected countries with a close relative in the US, such as a spouse, parent, child or sibling, will potentially be allowed in.
In a last minute change, the Trump administration extended the definition of close family to include fiancés.
However grandparents, aunts, uncles, nephews and nieces are not considered to be “bona fide” relations.
Moments before the ban began at 20:00 Washington time on June 29, it emerged that the state of Hawaii had asked a federal judge for clarification.
Hawaii’s attorney general has argued that the definition of “close family” is too narrow and may improperly prevent people from travelling to the US.
The American Civil Liberties Union (ACLU) said it would be closely monitoring the guidance and implementation of the order.
The Supreme Court is expected to make a final decision on the ban when its next session begins in October.
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