The most common and most likely answer to this crossword clue is the 6 letter word SHYEST
S
H
Y
E
S
T
More clues leading to the result SHYEST
Buddy explains!

The straight part
We know that "Supremely self-conscious" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case supremely self-conscious = shyest.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 SHYEST mean?
SHYEST adjective- Easily frightened; timid.
- Reserved; disinclined to familiar approach.
- Cautious; wary; suspicious.
- Short, insufficient or less than.
- Embarrassed.