Hinckley spent 11 years as a superintendent in Indianapolis' Warren Township schools and was praised for improving the district's financial status and test scores. She retired last summer.
She was the state's top superintendent while working in LaPorte before moving to Indianapolis.
White announced last month he would retire April 5 after nearly eight years at IPS's helm.
The state took control of three IPS high schools and one middle school this year and turned them over to private operators after years of poor student performance.