There are two main options for someone who wants to switch jobs in Saudi Arabia. First, you can ask your sponsor to transfer your sponsorship to the new employer or, you can ask your new employer to issue a new visa to you. Either way, will work.
Accordingly, this article aims to inform you about all the complications in this situation and to better guide you how to get a new visa (new company) after the Final Exit Visa (old/former company). It is assumed that you are already working in Saudi Arabia.