I think you mean EB status not E status, which refers to the priority level of the green card application.
I was really confused because an E3 is a visa very similar to a H1-B for Australians and an E1 is a visa for traders from countries that the US has a free trade agreement with.
The other course, depending on your country of citizenship, is asylum, which enables you to skip the queue and apply for your green card after a year. However, fraud is a big issue and you will eventually get slammed.