I don't know which city you are in. Let me tell you something about Beijing for your reference:
In Beijing, the new unit will generally ask for a copy of your medical blueprint or medical card (housing provident fund and other insurance can be handled by ID card), and then the unit manpower will go to the insurance institution or provident fund management center, so you don't have to worry about it.
I hope the advice is useful to you.
Further reading: How to buy insurance, which is good, and teach you how to avoid these "pits" of insurance.