回复：I have similar situation except 回复：H-1 Xfer - 回复：I also have

You may start to work for new employer any time after I-129 receipt has been issued, before your current H-1 expires.

So before I-129 transfer is approved, you may choose to work for either employer. Once the H-1 transfer has completed, and it becomes effective, you need to quit from currently employer and work for the new one.