After midnight, but generally before dawn(adjective)
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Use "postmidnight" in a sentence
"Mr. Bloomberg's postmidnight raid on the camp at Zuccotti Park swept the protesters from their spiritual home, but it re-energized the movement, too."
"I doubted that postmidnight snacks involving grilled fats and soda would do much for my waistline."
"There was probably no more poignant exit than Cal Ripken Jr. 's postmidnight departure in the rain from Baltimore's Camden Yards."