As in, complete the puzzle?
He was unable to piece the puzzle sooner.
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I've never seen piece used exactly that way before. I have, however, seen a common variation:
He was unable to piece together the puzzle sooner.
I don't believe that piece can be used as a verb on its own—but it can be used as part of a phrasal verb.
A similar example is:
He was unable to tease apart the puzzle sooner.
Here, although tease is a valid verb in its own right, tease is also used as part of a phrasal verb.