Excitation functions of elastic Si-28 + alpha scattering were measured employing the relatively new thick-target inverse kinematics technique, confirming the usefulness of the method. Data were taken at a Si-28 beam energy of 150MeV in a reaction chamber filled with He-4 gas, corresponding to the excitation region of 10-24MeV in S-32. Results are compared to earlier studies, in order to validate the method and explore its limitations. The data were treated in the framework of a simplified R-matrix approach. Energies, widths and spin assignments were obtained for 129 resonances in this mid-shell nucleus. The structure is discussed within the concept of a-cluster structure in the quasi-continuum of S-32.