2 H4 extensions may not work

 your child may file I-539 for H4 extension from June 1 to July 1. Then, he/she can file another H-4 extension together with your H-1 extension with the effective date on July 1.

Yes. There seems to be multiple options to fix the I-94 gap. I have been struggling to figure out the best way.

If file my son's H4 extension alone from June 1 to Junly 1, it may take 8 months to get approved. The H1B status cannot wait until my son's alone H4 is approved. So, will have to file H1B/H4 when my son's previous H4 extension is pending. Then my son will have two H4 extensions in parallel. Either one may get approved before the other. If the second H4 extension (which is with H1B) gets approved before the first, then the first alone extension's I-94 will be approved late and overide the alreay approved I-94 as the attorney told me, that means my son will not have a valid I-94 after July 1. There is always only one latest I-94 which will take effective and all others will become invalid.

Finally, I decided to renew my son's passport and fix the I-94 gap by a travel to China or Mexico. Though it is troublesome, but it is the safest way I know for now.

Thank you for the professional suggestions, Teacher 8!