The most common and most likely answer to this crossword clue is the 3 plus 10 letter word MAN MANAGEMENT
I've seen this crossword clue in The Daily Telegraph
M
A
N
M
A
N
A
G
E
M
E
N
T
Buddy explains!
The straight part
We know that "staff organisation" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case staff organisation = man management.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.