The most common and most likely answer to this crossword clue is the 6 plus 6 letter word LOUNGE LIZARD
I've seen this crossword clue in The Times
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More clues leading to the result LOUNGE LIZARD
Buddy explains!
The straight part
We know that "Monitor who comes in to seek coolness" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case monitor who comes in to seek coolness = lounge lizard.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.