Praise be to Allah.
It is not a necessary condition that the six days of Shawwal should be fasted consecutively. If you fast them separately or consecutively, it is OK. The sooner you do them, the better, because Allah says (interpretation of the meanings):
“…so compete in good deeds.” [al-Maidah 5:48]
“And march forth in the way (which leads to) forgiveness from your Lord.” [Al ‘Imran 3:133]
“[Musa peace be upon him said:] and I hastened to You, O my Lord, that You might be pleased.” [Ta-Ha 20:84]
And (you should hasten to fast these six days ) because delaying may cause problems. This is the view of the Shafi`is and some of the Hanbalis, but it is OK if you do not hasten it and you delay it until the middle or end of the month.
Al-Nawawi (may Allah have mercy on him) said:
“Our companions said: it is mustahabb to fast six days of Shawwal . Because of this hadith they said: it is mustahabb to fast these days consecutively at the beginning of Shawwal, but if one separates them or delays them until later in Shawwal, this is permissible, because he will still be following the general guidelines of the hadith. We have no dispute regarding this matter, and this was also the view of Ahmad and Dawud.” (Al-Majmu’ Sharh al-Muhadhdha)
And Allah knows best.