Jeffro
It is irrelevant to the subject of 537 BCE being the year of the Return as to whether May of 536 BCE can fall on in the second year of Cyrus. What is relevant to this matter is the events that are described in the account of Ezra 1:1-3:1. The simple fact of the matter is that Jews erected the altar with sacrifices in the seventh month in a unspecified year. Later in Ezra 3:8 the foundation of the temple was laid in the second year of their coming to the house of God. in the second month of that year. On the other hand, Josephus gives the second year of Cyrus for the laying of the Temple foundation. It would appear that Ezra is writing about the same event but from a different historical standpoint because the terminology used is not identical. This is a good example that when dealing with different historical sources, interpretation is required. Becausef the data for the same event differs it has no immediate relevance or bearing on determining the precise year when the Exiles returned for if this was the case then the whole matter would have been more clearly established in scholarship and to date this is not the case.
The fact of the matter is that it is not clearly stated the year of the returm but only the month which was the "seventh' month. We know that the Decree was given sometime in Cyrus' first year from Nisan 538 until Nisan 537 BCE so if Ezra used the Spring calender then the Returm must have occurred in the second year of Cyrus. But if Ezra used autumn calender then the seventh month would have fallen within the first year of Cyrus. So we cannot be certain what calender Ezra used in reference to the timing of the Return. Scholars and historians have considered all of these facts and there is a firm view that 538 BCE is highly unlikely because it does not give sufficient time for the Return journey by the Exiles so 537 BCE fits all of the facts nicely.
scholar JW