I received counsel that I needed to transfer from an E-3 to a H-1B before I could apply for a green card. That is what I did, and I now have a green card.

I also found that E-3s are sufficiently rare, I got conflicting comments from border agents, consular officials etc. Most had never heard of it, or applied understanding they had from other similar visas. (Admittedly, I first obtained mine the first year that category was created.)