The most common and most likely answer to this crossword clue is the 9 plus 8 letter word LANDSCAPE GARDENER
I've seen this crossword clue in The Times
L
A
N
D
S
C
A
P
E
G
A
R
D
E
N
E
R
Buddy explains!
The straight part
We know that "Outside design contractor" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case outside design contractor = landscape gardener.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.