(If anyone fasts during Ramadan, then follows it with six days in Shawwal, it will be like a perpetual fast.)
There is a social, religious and healthful indication in this instruction. Moving suddenly from Siyaam which requires refraining from food and drinks to eating freely will create a problem.
There should be a graduate movement from fasting 30 days to fasting 6 days whether successive six days or not, and this is the healthful indication. As for the social indication, your wife might break her Siyaam for few days in Ramadan because of a justified excuse, so if you observe Siyaam with her, she will be able to offer Siyaam of these days and your Siyaam will be mercifully supportive to her. As for the religious indication, fasting six days of Shawwah is Nafilah.