The most common and most likely answer to this crossword clue is the 9 letter word PANATELLA
I've seen this crossword clue in The Guardian
P
A
N
A
T
E
L
L
A
More clues leading to the result PANATELLA
- After a nap returned to order a cigar
- One smoker lights grass, displacing 1000 in country
- Smoke pot with a person counting votes, reportedly
- Headless insect found in dish that’s smoked
- What’s smoked oyster I’ve trimmed, included in Spanish dish?
- Old manand Fitzgerald generating smoke
- King Cole tucking into rice dish might enjoy this, postprandially
- Severely criticise mischievous goddess with everything going up in smoke
- Country, miles away, receiving order for cigar
- Criticise Americans accepting order for cigar
Buddy explains!

The straight part
We know that "might enjoy this, postprandially" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case might enjoy this, postprandially = panatella.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 PANATELLA mean?
PANATELLA noun- A long thin cigar.