It is highly recommended to fast six days of the month of Shawwal, as this is a meritorious act which results in abundant reward. However, it is not required to fast six days continuously without any interruption. One can fast according to convenience any time during the month.
In his response to the question in point, Sheikh Ahmad Kutty, a senior lecturer and Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states:

It is certainly wrong to suggest that one must rest for three days to celebrate `Eid Al-Fitr, since `Eid Al-Fitr is celebrated only one day. The six days of Shawwal fast can, therefore, begin right after `Eid, i.e., on the second day of Shawwal. As a matter of fact, that is the position of many scholars. The Prophet (peace and blessings be upon him) is reported to have said, “Whoever fasts the month of Ramadan, and then follows it up by (fasting) six days of Shawwal, it is as though he has fasted the whole year.” (Reported by Muslim)

However, there are other scholars who do not consider it absolutely necessary to begin the optional fasting of six days of Shawwal immediately after `Eid Al-Fitr. According to them, one may fast any of the six days of Shawwal. Neither do they consider it necessary for a person to fast the six days consecutively. So long as one has fasted six days either consecutively or separately, he has fulfilled the requirement and he will receive the same rewards.

Both of the above cited views have been transmitted from reliable authorities of Fiqh. It is, therefore, simply a matter of interpretation. Since there is nothing in the sources explicitly stating that it must be done immediately after `Eid or that it must be done consecutively, it must be judged as being flexible.

We must recognize the fact that taysir (ease and latitude) is integral to the nature of the Shari`ah, and, therefore, we are not supposed to make something rigid unless it has already been made so.