do I need to go to police to change my status when I finish my current job and wait for my new contract & COS

[hw620]

dear all
Hope you are all safe.
I have a question regarding the police registration.
My current job will finish on 30th Sep and I am waiting for the new employer to send me the contract and COS.
Do I need to go to the police station to update my status after 30th Sep or I can do it when I receive my new BRP?
I assume the police are not so clear about the immigration requirement, will they ask me to leave the country when I am waiting for the new COS?

Thank you for your help.