Killer left, drinking more rum? On the contrary (9)
I believe the answer is:
'killer' is the definition.
(I know that strangler is a type of killer)
'more rum? on the contrary' is the wordplay.
'more rum?' becomes 'stranger' (rum can mean strange).
'on' is an insertion indicator (as in clothing 'on' a person).
'the contrary' becomes 'l' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'stranger' enclosing 'l' is 'STRANGLER'.
'left drinking' is the link.
I am not very happy about this link. Some or all of it may belong to another bit of the clue.
(Other definitions for strangler that I've seen before include "Murderer" , "criminal" , "Killer compressing the windpipe" , "Killer by choking" , "he could take your breath away" .)