The most common and most likely answer to this crossword clue is the 2 plus 6 letter word ON TIPTOE
I've seen this crossword clue in The Times
O
N
T
I
P
T
O
E
More clues leading to the result ON TIPTOE
Buddy explains!

The straight part
We know that "Stealthily" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case stealthily = on tiptoe.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.