In striking down the other parts, lower court judges had cited Mr Trump’s campaign description of his policy as a “Muslim ban”.
Further arguments will be heard this week by federal courts in San Francisco, California, and Richmond, Virginia.
In June, the Supreme Court allowed an earlier version of the policy to take partial effect.
The president’s travel bans have each been frustrated by the courts to some degree:
In January, he signed an order banning people from seven Muslim-majority countries and suspending all refugee entry. The measure prompted protests and legal challenges across dozens of states.
A revised version in March exempted green card holders and dual citizens. By June, the Supreme Court allowed most of it to go into effect, a including 120-day ban on all refugees entering the US, but granted a wide exemption for those with a “bona fide connection” to the US.
That was superseded by Mr Trump’s third order, announced in late September, which added non-Muslim-majority nations North Korea and Venezuela