We expand the chromatic symmetric functions for Dyck paths of bounce number three in the elementary symmetric function basis using a combinatorial interpretation of the inverse of the Kostka matrix studied in Eğecioğlu-Remmel (1990). We prove that certain coefficients in this expansion are positive. We establish the $e$-positivity of an extended class of chromatic symmetric functions for Dyck paths of bounce number three beyond the "hook-shape" case of Cho-Huh (2019).