The inverse Kostka matrix transitions between Schur functions and complete homogeneous symmetric functions. Eğecioğlu and Remmel’s combinatorial interpretation for its coefficients uses objects called "special rim hooks" to decompose partition diagrams. We generalize their construction to the space of noncommutative symmetric functions (NSym) by first extending composition diagrams to include all integer sequences and then introducing "tunnel hook fillings" which decompose these diagrams. Statistics on tunnel hook fillings provide the coefficients for expanding the immaculate basis (a generalization of Schur functions to NSym introduced by Berg, Bergeron, Saliola, Serrano, and Zabrocki) into the complete homogeneous noncommutative symmetric function basis. We show how to use our formula to expand monomial quasisymmetric functions into dual immaculates, and further apply our formula to expand upon Campbell’s ribbon decomposition formulas.