Regardless of the time of day you decide to fly, the ticket price will remain around the same. Book flights from California to American Samoa 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 American Samoa. Flights to American Samoa from California in June are about $1,446 on average during June, but can be found for as low as $941.