Hat over umbrella any day.
First - because umbrellas are DANGEROUS if you are on an
ECV and moving, you could inadvertently injure someone else with the tips of the spokes that stick out from the umbrella. Not to mention that you then create an additional hazard because people can't easily see around you. You create a HUGE visual obstruction with an umbrella at seated ECV height.
Second - If you are on an ECV and holding an umbrella, you don't have both hands on the tiller. At WDW especially, you need to stay aware of what's going on around you, because people do have a rather alarming tendency to walk right into you, or right in front of you... I have never hit a pedestrian yet, but I have come far too close for my comfort. Kids are especially bad about darting out in front of you. Keep both hands on the tiller (the steering) for safety at all times!
Finally, if you require constant shade, you can rent
ECVs with sunshades, however I don't recommend it. I rented one with a sunshade the first time I used an outside vendor at WDW. Before the end of the second day, I had called and asked them to remove it. It limited my visibility when I needed to back up, it caused no end of problems on the buses, and more than half the time I was in the sun anyway, because the sun DOESN'T STAY IN A FIXED POSITION ALL DAY.

The sunshade is really only effective for a couple hours before and after noontime. After that, it's just a pain in the neck.
I have to wear a hat, and UV gloves (and long sleeves, and long pants) even at home when I leave the house. I have PMLE, and I will have a horrible burn with blisters, break out in hives, and have horrible flu-like symptoms if I am in direct sunlight for any length of time at all. My family jokingly calls me the Queen of the Mole People, because I do most of my shopping and errands after dark if I can. I can even get "burned" from simply being a room with too many unprotected windows.
At WDW, I gear up with hats, light colored UV protective clothing, and a touring plan that keeps me out of the sun during the midday when shadows are the shortest.