Character making final appearance in home game (5)

I believe the answer is:

omega

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'character' is the definition.
(letter of the Greek alphabet)

'in home game' is the wordplay.
'in' indicates the answer is hidden within the clue.
'OMEGA' is hidden within 'home game'.

'making final appearance' is the link.
This may not be right. Some or all of it may be part of another bit of the clue.

Can you help me to learn more?

(Other definitions for omega that I've seen before include "Last (Gk. letter)" , "Greek long 'o'" , "Final Greek letter" , "Final outcome" , "Foreign character" .)

I've seen this clue in The Times.

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