The most common and most likely answer to this crossword clue is the 6 letter word SMUGLY
I've seen this crossword clue in The Daily Telegraph
S
M
U
G
L
Y
More clues leading to the result SMUGLY
Buddy explains!

The straight part
We know that "with satisfaction" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case with satisfaction = smugly.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 SMUGLY mean?
SMUGLY adverb- In a self-satisfied manner.