Fasting out of the month Ramadan is extra (nafilah) worship. The rule of extra worships in Islamic Law is mustahabb. Shawwal is no different than the other months. There is no special extra reward (sawab) for fasting in Shawwal. The advices of our Prophet for fasting in Shawwal is just because it is a preferable time to encourage Muslims for extra fasting for that it is easier when you get used to it right after Ramadan. The reason of his advice being 6 days is that 36 days of fasting comes to a reward of 360 days in the measures of Quran and it was an effective way of encouraging people to tell that “6 days more from Shawwal will result in a reward for one year of fasting”. It is true, because Allah says:
“Whoever comes with a goodness will gain ten times its equivalent. Whoever comes with an evil will be requited with its equivalent. No one will be wronged.” (al-An’am 6:160)
So if you want to fast 6 days more after Ramadan, the one thing you have to do is to not fast on the first day of Shawwal, as it is Eid al-Fitr. There is no other necessity and no specific time for the extra worship.
Please see the link below for further detail: