The most common and most likely answer to this crossword clue is the 4 plus 1 plus 3 letter word HAVE A FIT
I've seen this crossword clue in The Guardian
H
A
V
E
A
F
I
T
Buddy explains!
The straight part
We know that "Almost lose it/as one will with last jigsaw piece" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case almost lose it/as one will with last jigsaw piece = have a fit.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.