As a follow-up to this question, Peter, is there a need to wait for the H1B VISA to be over or could the employer start the GC process right away?

They can definitely start it right away. There is no need to wait.

In fact, it's better to start the GC process at least 1-2 years before the 6-year max-out date to account for any delays or problems in the process.

Also A general followup. What is the recommended path to GC for a current holder of an H1B? Full 6 years first, or try and find I-140 immediately? another path?

Definitely start it right away. In fact, my recruiter friends mentioned that some companies are unwilling to hire H1B workers who have no GC sponsorship started and less than 2 years left in total.

So I would need to find a company willing to do that for me? It seems like where I'm at now is not willing to work on GCs.