You cannot look any earlier than 60 days, that's the general in-advance posting maximum for apartment listings in Jersey. You can start looking in June at the earliest for an August move-in. You can look in May to get an idea but nothing will be available for August that early. Since its a summer move-in its gonna be expensive and units disappear in hours. I don't know if any month is "cheaper" but probably looking in August is slightly better than in June.