Rich fabric not entirely satisfying (5)

I believe the answer is:

satin

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'rich fabric' is the definition.
'satin' can be an answer for 'fabric' (type of fabric). I am not certain of the 'rich' bit.

I don't understand how the remainder of the clue works.

Can you help me to learn more?

(Other definitions for satin that I've seen before include "Stain on shiny fabric" , "Saint (anag.)" , "Silky material" , "Lustrous fabric" , "Smooth and silky" .)

I've seen this clue in The Times.

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