You will need to transfer your H-1B visa to your new employer.
To do this, your new employer will have to go through the same process as your original employer, by filing a Labor Condition Application with the Department of Labor and then filing Form I-129 with USCIS. You can start working for the new employer as soon as the new employer files Form I-129 petition to transfer your H-1B visa. You don’t need to wait until USCIS approves the petition.