While H4 pending, may I apply a new H1B? Thank you so much for f

I had a H1B visa (expiration date 5-28-06, and not renewable) right now and a few days ago I submit a H4 application to keep my status. Now I received an offer (academic job, no H1 cap). My question is: can I apply the new H1 using H4 pending status? If not, how should I proceed from here? Thank you so much for sharing your experience and expertise.