The most common and most likely answer to this crossword clue is the 8 letter word STALKING
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More clues leading to the result STALKING
Buddy explains!

The straight part
We know that "approaching stealthily" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case approaching stealthily = stalking.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 STALKING mean?
STALKING verb- To approach slowly and quietly in order not to be discovered when getting closer.
- To (try to) follow or contact someone constantly, often resulting in harassment.Wp
- To walk slowly and cautiously; to walk in a stealthy, noiseless manner.
- To walk behind something, such as a screen, for the purpose of approaching game; to proceed under cover.
- To walk haughtily.
- The act of going stealthily.
- Hunting for game by moving silently and stealthily or by waiting in ambush.
- The crime of following or harassing another person, causing him or her to fear death or injury.
- The removal of stalks from bunches of grapes prior to winemaking.