Hawaii Association of Independent Schools [HAIS]
This link will take you to a list of private schools in HI. If you have time and resources to apply to and visit some of these schools, I would highly recommend looking into it. Compare salary/benefits to working environment. Talk with teachers already in the field. Positions with the DOE are more readily available, and will probably pay more. As a former teacher, I encourage you to explore your options carefully. Talk with DOE about the hiring process. You will also probably want to choose your island before applying. Visit if you can!