The most common and most likely answer to this crossword clue is the 9 letter word SWEATBAND
S
W
E
A
T
B
A
N
D
More clues leading to the result SWEATBAND
Buddy explains!

The straight part
We know that "part of a hat" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case part of a hat = sweatband.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does SWEATBAND mean?
SWEATBAND noun- A band of fabric, inside the crown of a hat, designed to absorb perspiration.
- A band of fabric worn around the wrist or head during sports to absorb perspiration.