The most common and most likely answer to this crossword clue is the 9 letter word GILLESPIE
I've seen this crossword clue in The Guardian
G
I
L
L
E
S
P
I
E
More clues leading to the result GILLESPIE
Buddy explains!

The straight part
We know that "might a cat have followed him?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case might a cat have followed him? = gillespie.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.