The most common and most likely answer to this crossword clue is the 4 plus 3 letter word GONE OUT
I've seen this crossword clue in Financial Times
G
O
N
E
O
U
T
More clues leading to the result GONE OUT
Buddy explains!
The straight part
We know that "On The Town possibly/dated" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case on the town possibly/dated = gone out.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.