We in Nepal have a similar problem since they is not variation to airline job as is in the US. However, the trend has been that the pilot with CPL gets a job as a FO, and the wait can be as long as 5 years. And people have waited, it is hard coz you cannot get a job elsewhere, visa restriction, license conversion, lack of hours.
However, where your FAA license can come handy and you can work as CFI in areas where FAA is being used like in Guam, Israel is one of them, etc. Being a Singaporean, I am sure you'd have not much problem, visa is much easier. But besides this, I am sure you can look into working in Malaysia, since both the countries has some kinda ties. Malaysia does has GA. But a lot of people to build hours do end up in Africa personally and try to get a job to fly the single engines etc.
Also, a private air charter company in singapore that does corporate work etc. should be an option right?
No, Malaysia has very strict regulations protecting their own pilots.