Regardless of the time of day you decide to fly, the ticket price will remain around the same. Book flights from Ithaca to New Hampshire as you normally would.
Due to multiple factors, expect to potentially pay more when booking in January. For low rates, June can be a great choice for travel to New Hampshire. Flights to New Hampshire from Ithaca in June are about $294 on average during June, but can be found for as low as $147.