After some analysis, it's probably a Tudor bonnet.
This is the second consecutive
Last time, Senator Claire McCaskill (D-MO) tweeted about the justice's "weird Hat."
The particular genus of that hat has remained unidentified for four years.
Tudor bonnets are traditional academic dress for a person who holds a professional doctoral degree. It's made of velvet or cloth.It's unclear whether Scalia's hat is in fact a Tudor bonnet, but it bears a distinct resemblance. It also coincides with Scalia's J.D. from Harvard Law School.
Any haberdashers with additional insight on the headgear are encouraged to respond with additional information.
UPDATE: A reader has sent along this article which identifies Justice Antonin Scalia's head gear definitively. According to Kevin Walsh, a professor at the University of Richmond School Of Law:
The hat is a custom-made replica of the hat depicted in Holbein’s famous portrait of St. Thomas More. It was a gift from the St. Thomas More Society of Richmond, Virginia. We presented it to him in November 2010 as a memento of his participation in our 27th annual Red Mass and dinner
More lived during the Tudor dynasty so it is likely that the hat is of the Tudor Bonnet variety.