Use 'rush out' when describing someone leaving a place quickly and without warning. He rushed out as soon as he got the phone call.
Pro Tip 2/3
Emergency Context
'Rush out' is commonly used to describe actions during emergencies or urgent situations. The athletes rushed out of the stadium after the announcement.
Pro Tip 3/3
Emotional Nuance
When used figuratively, 'rush out' can imply an emotional reaction, like leaving in anger or distress. After the argument, she rushed out in tears.