The most common and most likely answer to this crossword clue is the 5 plus 6 letter word URBAN LEGEND
I've seen this crossword clue in The Times
U
R
B
A
N
L
E
G
E
N
D
More clues leading to the result URBAN LEGEND
Buddy explains!

The straight part
We know that "Second-hand story" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case second-hand story = urban legend.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.