What would cause a person to have headaches from the sun only when not wearing a hat?

There are 4 main reasons, which we discuss in detail below:

Reduced tolerance for light

Greater sunlight sensitivity between attacks

Longer duration of sun exposure

Exposure to specific wavelengths emitted by sun

People prone to light-induced migraine episodes generally have a lower threshold for light. In fact, the light of an overcast and cloudy day can be enough to cause pain! Thus, even normal levels of light exposure—much less bright days—can lead to headaches and other migraine symptoms.
The tolerance for light can also be lower between attacks making people more sensitive even when they are not in pain. This means a person may not immediately have a headache while outside, but it may be building. And if you just came out of an attack, you may find that your symptoms linger as a result of sun exposure.
If you don’t have an immediate headache or migraine from the sun, experts have further suggested that the cumulative effect of sunlight exposure over time can be just as damaging. Ultimately, the longer you stay outside, the more likely you may develop a headache.

Lastly, the sun emits a spectrum of light wavelengths, one of the strongest of these being high energy visible light or what most people label blue light. In fact, blue light is everywhere—fluorescents, device screens, and other artificial sources as well. And blue light has been repeatedly identified as the most painful color of light for people with migraine. thus the combined effect of sun brightness and these painful wavelengths can be a dynamic duo of unpleasantness.