Island Air closing could have some influence, but it's not much. It all really started back when Aloha Airlines shut down in 2008. With no major competition interisland, Hawaiian can name it's price. And it's even worse with smaller airports such as Hilo, that has absolutely no other interisland competition (the other island airports have Mokulele, a smaller interisland airline service).
Southwest is suppose to enter, but only between West Coast to Hawaii ... I would welcome them to interisland flying if it can help bring prices down for interisland travel.
In Oahu, there is actually one other smaller airport that people can choose to fly into .. but for me, I only fly into the HNL airport since it's the major one.