Small vessel put under the drink (7)

I believe the answer is:

coaster

I'm a little stuck... Click here to teach me more about this clue!

'small vessel' is the definition.
'coaster' can be an answer for 'vessel' (I've seen this before). I am not sure about the 'small' bit.

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

Can you help me to learn more?

(Other definitions for coaster that I've seen before include "proceeds with little effort" , "on which some mug may be made to stand?" , "... another one [VESSEL]" , "Ship; (drinks) mat" , "Mat to protect a table" .)

I've seen this clue in the Sydney Morning Herald.

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