Of course, since there were only 24 orders, that meant that each served twice a year on opposite ends. So Zachariah would have also served about six weeks after Sukkot before returning, potentially pushing John's birth to early- or mid-winter.
The reason I believe the first option has to do with the significances of the Feasts. It makes far more sense for John to have been born at Passover (when expectations for the coming of Elijah are at their highest) and Yeshua on Sukkot (when we celebrate the building of the Tabernacle and Temple and God dwelling with His people) than the other way around.
Shalom!
I'm just joyful that Jesus came to save my life. Hallelujah.