The Spanish Sombrero comes from the Latin prefix sub– (“under”) with the Latin umbra (“shade, shadow.”) We also get the Spanish sombra (“shade”) from that same root, as well.
From that same root umbra, we get the English… umbrella. The umbrella does protect you from the sun, actually — think of 19th century aristocratic women walking around with their sunlight umbrellas!
We also get, from the same root, the English to take umbrage — to get angry. Why? The bad things that happened that got you angry usually happened… in the shadows.
The u-mbr root is clearly visible in all the variations.