The chances of getting HIV infection from the risk exposure you describe are very low, therefore I would not worry too much. However there is a risk, however small, and therefore I think you have made the correct decision to get another test at 3 months or later.
Any of the large hospitals in the public or private sector should be able to provide a confidential test. If you are concerned, give any name and ID data, not necessarily your correct name. There are also Voluntary Testing and counselling Centres (VTC) in Bombay.