Travelling from most locations, it is best to book your flight to Kagoshima during the month of June for the best deal, although there are other times of the year which offer great deals as well.
In general, booking your departure to Kagoshima for a Wednesday represents the best deal on airfare. Try to avoid a Sunday departure to stay away from the highest prices.
Typically, cheap prices to Kagoshima are most often found for morning flights. Cheap prices are less likely for midday flights.