Drink to follow for small pet (6)

I believe the answer is:

lapdog

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

'drink to follow' is the wordplay.
'drink' becomes 'lap' (lap can mean to drink).
'to follow' becomes 'dog' (dog can mean to follow or torment).
'lap'+'dog'='LAPDOG'

'for' acts as a link.

Can you help me to learn more?

(Other definitions for lapdog that I've seen before include "One docile and subservient" , "one of man's more diminutive best friends" , "One readily submitting" , "lackey" , "Pom, for one" .)

I've seen this clue in The Times.

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