Shoulda had a transparent bill to the cap, right?

I rarely wear a baseball cap while flying--often while driving, but not flying. Part of the reason is that most baseball caps and headset bands are incompatible, but part of the reason is that it does limit visibility upwards.
The fact is this, though: "see and avoid" doesn't really work very well. Sure, there are those instances in which a pilot was too involved looking inside the cockpit when he/should have been looking outside, but a good many mid-airs which have occurred involve pilots who were looking but didn't see--things got in the way. Most of the airplanes we fly have lots of blind spots.
Cary