You need to keep all government agencies up to date re your actual address, including the Seguridad Social, as a change of address might involve a change of medical centre etc. as well and they keep records of all that, e.g. to facilitate online ´cita previa´ facilities that will automatically select the GP you´re registered with, if your data are correct. If you move from one autonomous region to another, they might need to issue a new card.
A new, national medical card has been introduced, but it will be a couple of years before all existing cards will have been exchanged.
You can inform the Seguridad Social in person, or through their online office, with either a digital certificate installed on your computer, or with a Cl@ve PIN. Info on both in our Library, Certificates section and Taxation section. You will need THIS LINK and scroll down to ´change of address´, click and choose one of the above mentioned electronic ID methods. The actual change menu is in Spanish, click ´continuar´ till you reach the end. Don´t use any punctuation marks, just letters in the address box, numbers in the box for housenumber.
If you have moved to another autonomous region, you will then need to go to the website of the Junta of your region, to apply for a new card. Or make enquiries at your local health centre if they need to do that for you.
You can also inform them in person. Within the same town, you can just update your padrón, bring a padrón certificate and go to your local medical centre to have your address updated.
If you move from one town to the other, but within the same Autonomous region, the above mentioned applies as well, but if you move to another Autonomous region, you need to visit the INSS office nearest to your new town, with your ID, old card and padrón certificate. Would bring Residency Certificate as well, and if pensioner, your S1 form.
As not all databases for all the regions are connected, they might need some time to verify your entitlement with the INSS in your previous region. In practice, they will issue a temporary entitlement certificate and you will either receive new cards by mail, or you can go to your new local medical centre to apply for them after a couple of week. This differs per region, so best to check when you´re at the INSS office.