Wake wanting bit of medicine at hospital (9)

I believe the answer is:

aftermath

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'wake' is the definition.
(I know that wake can be written as aftermath)

'wanting bit of medicine at hospital' is the wordplay.
'wanting' becomes 'after' (I can't justify this - if you can you should believe this answer much more).
'bit of' suggests taking the first letters.
'hospital' becomes 'H' (abbreviation for hospital).
The first letter of 'medicine' is 'm'.
'after'+'m'+'at'+'h'='AFTERMATH'

Can you help me to learn more?

(Other definitions for aftermath that I've seen before include "Frame that in following events" , "Subsequent circumstances" , "What may be the later consequences" , "Situation following as consequence or result" , "Circumstances that follow as a result of something" .)

I've seen this clue in The Guardian.

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