The most common and most likely answer to this crossword clue is the 6 letter word TAUTER
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Buddy explains!

The straight part
We know that "increasingly stretched" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case increasingly stretched = tauter.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 TAUTER mean?
TAUTER adjective- Under tension, like a stretched bowstring, rope, or sail; tight.
- (of a body, muscles, etc.) Not flabby; firm, toned; (of a person) having a lean, strong body.
- (of music, writing, etc.) Containing only relevant parts; brief and controlled.
- Experiencing anxiety or stress.
- (of a sailor, a ship, etc.) Neat and well-disciplined; (by extension) efficient and in order.
- Strong; uncompromising.