How can I expedite my application process to get the blue card in Greece?

Due to the prevailing circumstances, many temporary residence permits facing expiration should be arranged through automatic renewal by the re-opening date from the decentralized authorities. But if you have provided reasons to expedite the procedure, you can submit a letter (or send an email) explaining the reasons your permanent residence permit must be issued immediately; attach any supporting documents too, if available.
The request is then examined by the head of the decentralized authorities and, if accepted, within a month, the biometric residence permit is issued.

You will have to apply for renewal for one more year at the immigration office.