I want to know if I could start fasting the 6 days of Shawal before or after making up the days I missed fasting in Ramadan?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

The best thing to do for a person who has to make up his Ramadan fast, is to fast the missed days of Ramadan before starting supererogatory fast because it is preferable to perform his obligation as early as possible. Imam Ahmad said: “It is not permissible for one to fast non-obligatory fast before making up the missed obligatory fast”.

Allah knows best.