The most common and most likely answer to this crossword clue is the 11 letter word NECKERCHIEF
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More clues leading to the result NECKERCHIEF
Buddy explains!

The straight part
We know that "protection for head’s supporter" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case protection for head’s supporter = neckerchief.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 NECKERCHIEF mean?
NECKERCHIEF noun- A scarf that is worn looped or tied around the neck.