Actually, that's not how it works! Don't pay too much attention to systems that try to tell you otherwise.
Your hair color doesn't make a darn bit of difference in determining your season. A person with red hair might be found in any season, depending on her skin tone.
Here's a redheaded Light Spring:
A redheaded True Spring:
A redheaded True Autumn:
And I personally know a redheaded True Summer.
Redheads tend to have warm-toned skin, which puts them into Spring and Autumn categories. But it's theoretically possible that a person with genetically red hair could have Winter or Summer skin.
Unsure of your season? Try at-home draping cards.