May come out of bed to rebuild a storey (6)

I believe the answer is:

oyster

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'may come out of bed' is the definition.
I can't tell whether this defines the answer.

'rebuild a storey' is the wordplay.
'rebuild' is an anagram indicator (I've seen this in other clues).
'storey' is an anagram of 'OYSTER'.

'to' acts as a link.

Can you help me to learn more?

(Other definitions for oyster that I've seen before include "Parent of the pearl" , "Pearl grower" , "Pearl-yielding shellfish" , "Pearl creator" , "Bivalve native" .)

I've seen this clue in the Evening Standard.

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