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, because Allaah says (interpretation of the meanings):
so compete in good deeds al-Maaidah 5:48
And march forth in the way (which leads to) forgiveness from your Lord Aal Imran 3:133
Moosa 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 Shaafa 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 Allaah have mercy on him) said:
Our companions said: it is mustahabb to fast six days of Shawwaal. Because of this hadeeth they said: it is mustahabb to fast these days consecutively at the beginning of Shawwaal, but if one separates them or delays them until after Shawwaal, this is permissible, because he will still be following the general guidelines of the hadeeth. We have no dispute regarding this matter, and this was also the view of Ahmad and Dawood.
Al-Majmoo Sharh al-Muhadhdhab
Sheikh Muhammed Salih Al-Munajjid