If you have an excuse that keeps you from fasting the six days of Shawwal in Shawwal because you have to make up missed fasts, then you may fast six days of Shawwal in Dhul-Qa'dah

`Eid Al-Fitr is just one day on which it is prohibited to fast. As for the second and third days of Shawwal, it is not prohibited to fast on those days. It is permissible to fast it to make up a missed Ramadan fast or as a voluntary fast.

There is nothing wrong with making up a missed Ramadan fast in the last half of Sha’ban.

It is permissible for you to make up missed Ramadan fasts on the day of doubt and one or two days before Ramadan. For more, please see the detailed answer.

If you start an obligatory fast, such as making up a missed Ramadan fast or offering expiation for breaking a vow, it is not permissible for you to break the fast without a valid excuse, such as sickness or travel.

It is not essential to make up for missed fasts consecutively. If he does that, it is better, but if he does not there is nothing wrong with that.

The majority of scholars said that it is not permissible to fast on the Days of Tashriq whether it is an optional fast, making up a missed fast or fasting in fulfillment of a vow. So, the fasts that you observed on the Days of Tashriq to make up for missed Ramadan fasts, are not valid, and you must repeat them.

Delaying making up missed fasts is a sin, so you have to: 1- repent to Allah, 2- hasten to fast the number of days you think you missed, and 3- feed one poor person for each day if you can afford to do that.