The most common and most likely answer to this crossword clue is the 13 letter word STRIPPERGRAMS
I've seen this crossword clue in Financial Times
S
T
R
I
P
P
E
R
G
R
A
M
S
Buddy explains!
The straight part
We know that "Hired messengers who become increasingly exposed" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case hired messengers who become increasingly exposed = strippergrams.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.
What does STRIPPERGRAMS mean?
STRIPPERGRAMS noun- A stripper hired to deliver a message or greeting.