Shanghai Social Security Card Used in Jiaxing

I can't. Jiaxing belongs to Zhejiang, and it is inter-provincial with Shanghai, so it can't be used in Jiaxing. Unless you are hospitalized, even so, you need to issue a referral certificate in the designated hospital of Shanghai medical insurance for referral filing, so that you can immediately settle the medical insurance in different places when you are hospitalized in Jiaxing.

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