A hat that has a hard straight brim that is a tefach wide is prohibited to be worn on Shabbos as it is making a tent.

Chapter 301 Section 41

When you are wearing a hat on top of a yarmulka, you must be careful that the hat fits on tight so that the wind will not blow it away, as you might come to carrying.

Chapter 301 Section 42

If you find a kosher pair of tefillin in middle of the street on Shabbos, you can put them on and wear them until you get home.