There are four valid sayings of our scholars regarding this. And the easiest for the Muslim should be followed. Meanwhile, the fourth saying is argued weak.
1: If someone has missed some of the days of Ramadhaan, he should fast them first, then fast six days of Shawwal, because he cannot follow the fast of Ramadhaan with six days of Shawwal unless he has completed his Ramadhan fast.
2: If there are too many, and it will push you over and you will miss Shawwal, don’t worry. As soon as you are through with the qadaau (missed Ramadhaan), then fast the “six days of Shawwal” whatever the date is, and you will get the reward for it In Shaa Allah; this is based on the necessity.
3: If you think you will find it difficult to make up all your missed obligatory fasts now, especially because of the length of day and heat, then leave them for later in the year and fast the six days of Shawwal now. This was performed by ‘A’ishah and despite the discussion over her action, her example can be used as an evidence for anyone who cannot fast the obligatory qadau(repayment) now; and will put the voluntary fasts before the obligatory.
4: Whoever fasts in repayment of the missed Ramadhaan simultaneously with the six days of Shawwal have fulfilled their religious obligation as well as obtained the reward for fasting the six days of Shawwal.
Scholars of the Shaafi’iyah permitted this. Making a repayment(six fasts or above) of the Ramadhaan in the month of Shawwal will benefit from the reward of fasting Ramadhaan and six of Shawwal.