I'm not quite sure what your question is. Since you already seem to have an API that returns dates and you have already hooked up a dropdown then I assume the issue isn't with getting the data/updating UI but rather just getting the 6 months of dates. For that just add 6 to the month number.
public static IEnumerable<DateTime> GetNextMonths ( DateTime startDate, int months )
{
for (var index = 1; index <= months; ++index)
yield return startDate.AddMonths(index);
}
var next6Months = GetNextMonths(startDate, 6);
Assumption here is that you want exactly 6 months, bearing in mind that some dates (like 1/31) would not properly translate 1 month but that is probably not an issue.
If you want a slightly more elegant, but properly less efficient approach then.
var next6Months = Enumerable.Range(1, 6).Select(x => startDate.AddMonths(x));