Dante Alighieri's hat

The white cap is a linen coif. The red cap might be a sock hat? Or a simple Chaperon that's not been wrapped up? Hard to tell from the photos of the paintings.

There's a whole host of Renaissance men's headgear...

The coif is the white linen skullcap worn under all headgear.

That's no chaperon. A chaperon is a huge confection, but its ancestor is a simple hood with a long point (a "liripipe") in back and a dagged shoulder cape -- except that the face opening goes over the top of the head instead of around the face and the dagged cape and long point hang down arranged in various ways.

Some clever dandy thought of it, and it became its own stylized headgear with a padded doughnut around the crown which used to be the face opening of a simple hood, and a little scalloped scarf to one side which used to be the shoulder cape and a long tube of what used to be the liripipe draped over a shoulder.

There are no depictions of Dante from his own lifetime. All later depictions show him with a white coif and a simple red pointed cap with the point a little droopy and the brim turned up. I suspect it was a Renaissance idea of "simple headgear of the remote past," since his garments are also usually depicted as vaguely classical and of no particular time.