Rug, cheaper one I tossed out (9)

I believe the answer is:

hairpiece

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'rug' is the definition.
(I've seen this before)

I can't explain the rest of the clue.

(Other definitions for hairpiece that I've seen before include "rug" , "Using switch" , "False addition to existing growth on the head" , "A wig or switch to supplement scalp covering" , "Switch, for instance" .)

I've seen this clue in The Times.

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