Government in Peconic, NY Government Offices City Southold TownPECONIC LnPeconic, NY, 119586317345211Government CityGovernment Offices City Town Of BrookhavenSouthold Town Senior ServicesUnemployment Office Southold Animal Shelter269 Peconic LnPeconic, NY, 119586317651811Animal Rescue SheltersGovernment City Village TownshipGovernment OfficesTown OfUnemployment Office Show MoreSee AlsoGovernment OfficesCity, Village & Township Government