The structures of four types of amorphous silicon are examined by an experimentally constrained structural relaxation method (ECSR). Experimental selected area electron diffraction data and fluctuation electron microscopy normalized diffraction variance data were used as constraints to guide a Monte Carlo relaxation procedure towards best fit models. A Tersoff potential was also used to further restrict the space of possible solutions. The materials examined were self-ion-implanted silicon and pressure-amorphized silicon, both in their as-prepared and thermally annealed states. In the fitted models for these materials regions containing two types of medium-range order were identified. One type involves formation of paracrystallites with cubic and hexagonal structures, where both short-range crystalline and medium-range order are present. The other type of medium-range order appears in the form of extended crystalline planes without associated short-range crystalline order. These two types can coexist. It is observed that the best fit models for both as-prepared samples contain approximately 10-15 paracrystalline ordered regions, reducing to about 5-10 in the annealed materials. None of the models are true continuous random networks. We conclude that, with long computational times and with a suitable potential function, the ECSR procedure provides a powerful, although at present semi-quantitative, tool for determining the structural form of medium-range order in thin amorphous materials.