The most common and most likely answer to this crossword clue is the 4 plus 2 plus 3 plus 4 letter word LOOK UP AND DOWN
I've seen this crossword clue in Financial Times
L
O
O
K
U
P
A
N
D
D
O
W
N
Buddy explains!
The straight part
We know that "Search thoroughly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case search thoroughly = look up and down.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.