(Hekatombaion 1) Oddly, Athens had no real festival to mark the beginning of the year, although magistrates and councils made sacrifices to Zeus the savior (of the city) and to Athena the savior on the last day of the previous month. They were seeking protection of the city for the coming year (Parke, p.29). The new archon was sworn in at the New Year and his first act was to proclaim that everything everyone owned would continue to be theirs (Aristotle, the Athenian Constitution, 56.2). All business from the previous year had to be completed before the year ended: even criminal trials could not be carried over into the New Year (Burkert, p. 228).
According to Burkert, the Athenians began to conclude the year as much as two months before the Panathenaia festival. He noted that the purification of the central sanctuary of Athena Polias was undertaken, beautifying and washing (at the Kallynteria and Plynteria), cleaning away the old to make way for the new. At these rituals, “death and primitive times are recalled, but at the same time a new beginning with cultivated food is anticipated” (p. 228). It’s also worth noting that the New Year in Athens was observed during the “slack” period agriculturally in the region, right after the grain harvest. Although the major celebration of the New Year occurred at the Panathenaic festival later in the month (on what was considered the “birthday” of the city), one of the major practices to welcome the New Year occurred at the monthly sacrifice to Apollo, on the 7th, when a hekatombaion of oxen were sacrificed. This ancient practice had given the month its name (Burkert, p. 231).
If you wish, you could make a small offering to Zeus the savior and Athena the savior on behalf of your city or county the day before the first of Hekatombaion.
Burkert, Walter, Greek Religion, 1977, English version 1985
Parke, H.W., Festivals of the Athenians, 1977