Depending on the model of your Garmin GPS unit... not all models are supported by Garmin Singapore. Also, you are required to pay to download of maps for countries other an Singapore and Malaysia.
First find out whether the US car-rental company provide cars with GPS? Next best, make a trip to Sim Lim and look for Garmin Set sellers and ask whether they can provide the map for a small fee. Online download would cost cost around USD90