Can STEM F1 student with approved H1B change job before 1 Oct 2013?

I am on STEM F1 working for E-verify company. Recently my H1B got approved. I am offered job at another company. Another company is willing to apply for H1B transfer so that I can work for them. If H1B transfer gets approved, can I start working for another company before 1st Oct? This another company is not e-verified.

If you are on OPT extension, then you cannot work for a non e-verified company even after H-1 gets approved. You can work for non e-verified company after Oct 1 once you move to H-1 status.

This means, you continue to work for current employer until Oct 1. Have the new employer file cap-exempt H-1 petition for you (aka H-1 transfer) w/ start date of Oct 1 and join them from that date.