When wool is dyed before being spun into thread (as opposed to after it is spun or woven into fabric), the color is profound and likely to last a very long time. From this we can infer the metaphorical meaning of the idiom dyed in the wool, which means profoundly, deeply ingrained, or to an extreme degree. It’s usually used in describing a person’s political, cultural, or religious beliefs or to emphasize their commitment to something.
The phrase is usually hyphenated, especially when it comes before what it modifies (e.g., he is a dyed-in-the-wool Yankees fanatic), but it can go unhyphenated when it comes after what it modifies (e.g., as a Yankees fanatic, he is dyed in the wool).
Died in the wool appears occasionally. Sometimes it’s a pun, but it’s usually just a misspelling.
Examples
[J]azz’s popular appeal has been weak in recent years, and dyed-in-the-wool “jazz police” purists haven’t helped. [National Post]
Unique is an overworked word, but Anderson is just that, a dyed-in-the-wool stylist whose eccentricities are as distinctive as fingerprints. [Milwaukee Journal Sentinel]
But the dyed-in-the-wool Light Blues fanatic won’t be able to forgive and forget if his beloved club are found guilty of financial crimes. [Scottish Daily Record]
Johnson’s lengthy courtship of dyed-in-the-wool Southern racists like Richard Russell and Harry Byrd allowed him to learn the Senate’s bylaws. [The National]
I’ll be among those dyed-in-the-wool Royal enthusiasts waving my flag in front of Her Majesty herself outside the gates of Buckingham Palace. [Manchester Evening News]