The most common and most likely answer to this crossword clue is the 7 letter word COLUMBO
C
O
L
U
M
B
O
Buddy explains!
The straight part
We know that "Lieutenant in light brown overcoat, initially exhibiting hesitation" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case lieutenant in light brown overcoat, initially exhibiting hesitation = columbo.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.