I think that it is a bit late to file H4.

I answer a similar question two days ago. You can file H4 and then send the receipt to the place where your H1 is pending. However, no guarantee the officer will receive it or consider it. The decision of your H1 may come before they receive your H4 receipt or the officer may just throw it away. The worst case, leave and come back on a H1 visa. On the other hand, the officer may overlook the GAP.