The most common and most likely answer to this crossword clue is the 9 letter word OVERPRINT
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More clues leading to the result OVERPRINT
Buddy explains!

The straight part
We know that "one's doubly impressed" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case one's doubly impressed = overprint.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 OVERPRINT mean?
OVERPRINT noun- The addition of new text on a previously printed stamp, usually to add a surcharge or change the face value.
- To print over what has already been printed.
- To add an overprint to (a stamp).
- To print too many copies of.
- To overlap different colours to avoid gaps.