I changed companies. Should I change my social security card?

Just handle the insurance transfer, there is no need to change the social security card.

The required materials are as follows:

You need the labor contract termination certificate and personnel reduction form of the original unit, issue a "social security payment voucher" at the social security center of the transfer place, and provide an insurance transfer form. Then take these and the endowment insurance manual to the human resources department of the new company, and they will submit the vouchers and applications to the social security center of the transfer place for review. If there is no problem. It is estimated that your social security will be transferred to the transfer place within 1-3 months.

Further reading: How to buy insurance, which is good, and teach you how to avoid these "pits" of insurance.