Abu Ayyub (May Allah be pleased with him) reported:
The Messenger of Allah (ﷺ) said, “He who observes As-Saum (the fasts) in the month of Ramadan, and also observes As-Saum for six days in the month of Shawwal, it is as if he has observed As-Saum for the whole year.” (Riyad as-Salihin, Book 9, Hadith 1254)
You can start fasting six days of Shawwal from the second day of Shawwal, because it is haram to FAST on the day of ‘Eid. You can FAST the six days at any time during Shawwal, although the best of good deeds are those which are done soonest.
It is not a necessary condition that they should be fasted consecutively. If you fast them separately or consecutively, it is OK. The sooner you do them, the better