Hong Kong Club Annexe [1903-c.1966]
I'm not sure of the correct name for this building, but on a 1911 map of Central (#3-4, mapping HK) it is marked as as 'Jockey Club Office. Hongkong Club Annex'.
Also in this photo you can see there was a covered walkway connecting it to the main Hongkong Club building:
Can anyone help fill in more details?