H1B Max out but no Perm filed

Hi,
My H1B Max outs Jun 2020. Employer has not filed PERM till now. Please suggest if I can do COS to H4 before H1B maxout to stay in USA?
Assuming a new employer files PERM for me while I am in H4 status after June, do I need to make sure h4 effective date is before H1B Maxout date to be cap exempt?

Thanks,
Saravana

You can file H4 COS before the max out.

Thank you Anil for the response.

Hi Anil,
I tried to file H1B to H4 COS online. But I could not see the ‘Current Immigration status’ drop down with H1B. Can you please let me know if this case cannot be filed online?
Thanks

If there is no option, then it may not be allowed to file online.