Can H1-B visa be transfered to a new employer before the actual October 1st begin date


I’m currently working on CPT and my H1 has been approved with my current employer, but the client has decided to scrap the project and so I won’t be able to work with the current client after Aug 30th.

Meanwhile I had been looking for full-time opportunities and I I’ve recieved a full-time offer from another employer, who wants me to join before October 1st. Also, my current employer has suggested to get another contract project and he will see through the change in LCA or an ammendment in the current visa, whichever applicable.

So I’m wondering if I can transfer my H1 to the new employer or do I need to stick with the current employer until I’ve 2 months of paystubs before my new employer can initiate the H1 transfer process?

What should be the right way to go, please shed some light, and thanks for taking the time out to answer.