It is difficult to figure out which day best corresponds with a date on the ancient Greek calendar. There are a few reasons for this including the fact that each polis might use a different calendar. This page on the festivals of ancient Greece explains some of the problems we face even with the calendar of ancient Athens:

"Athens tracked 3 different calendars, the Prytany Calendar which divided the year into ten parts during which each tribe rotated chairing the Assembly, the Festival Calendar by which actual dates for sacrifices, etc. were scheduled and was very loosely related to the actual phases of the moon, and the actual lunar cycle known as Kata Selenes, or Kata Theon (according to Selene, or to the Gods). Kata Selenes produces a very irregular pattern of 29 and 30 day months. Any attempt to follow a regular pattern of 29 and 30 days will quickly fall out of synch with the moon."

This Day in Ancient History
Greek Months