Trump defended the action even as his chief of staff walked back a controversial element.
Visa-holders and refugees arriving late Friday were detained; the travel ban also was applied to green-card holders.
The president’s action removed protections for personal data, alarming advocates.
Mark Morgan’s departure comes one day after President Trump’s crackdown on illegal immigration.
He will also need Congress to appropriate an estimated $20 billion.
The moves represent Trump’s first effort to deliver on perhaps his signature issue of immigration.
Retired Marine general is charged with helping securing the border with Mexico.
Kelly appeared Tuesday afternoon before a Senate committee considering his nomination.
Don McGhan, a well-known campaign finance lawyer, will be White House counsel; K.T. McFarland, a national security analyst, to be deputy national security adviser.