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You can fast the six days of Shawwal either immediately after the Eid day or delay them. You can also fast them either sequentially or separately throughout the month of Shawwal.
If you have some days of Ramadan you missed, you still can fast the six days of Shawal before them according to the majority of the scholars but Imam Ahmad said: ‘It is not permissible for one to fast non-obligatory fast before making up the missed obligatory fast.’
So the best act for a person who has to make up his/her Ramadan fast, is to fast the missing days of Ramadan before starting supererogatory fast, since it is preferable to perform his obligation as early as possible but there is no problem with you doing the 6 of Shawal first if time does not allow you to do both.