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In this paper, an approach is developed that allows one to substantially accelerate numerical solving of the eigenvalue modal and scattering problems treated with the aid of the mode-matching technique or integral-equation technique combined with the method of moments for a wide class of azimuthally inhomogeneous cylindrical structures having sector-shaped slots. It is based on a special decomposition of the matrix elements containing series over spatial harmonics. An earlier developed decomposition scheme is modified that provides fast and accurate calculation of the series, which correspond to both slot and nonslot domains. Two new efficient schemes are proposed, which are especially important for accelerating calculations in cases where the series corresponding to the slot domains contain cylindrical functions of noninteger order. Results are presented that demonstrate advantages of both the new and modified decomposition scheme.