Very close to the right rhyme (5)

I believe the answer is:

tight

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'very close' is the definition.
'tight' can be an answer for 'close' (synonyms). I'm unsure of the 'very' bit.

I can't explain the rest of the clue.

Can you help me to learn more?

(Other definitions for tight that I've seen before include "Squeezed" , "Unlikely to be choosing the present" , "reluctant to buy a round?" , "Drunk - under control" , "Mean - drunk" .)

I've seen this clue in the Evening Standard.

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