Get your approval notice, then ask the other company if they can file a petition
if it's approved, ask new employer to transfer it and switch company after the transfer is complete.
H1 can be transferred only after it starts, that is on October 1. Also lawyers suggest you have at least 2 pay cycles from current employer to avoid RFE.
Before October 1, its called filing amendment and not transfer. It has chances of rejection unlike transfer. Wait until first week of October if its jot end of the world !
Why do people not search for similar posts before asking? There are previous posts with comments addressing the exact same issue.
Not if you want your H1