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TiFe intermetallic compound has been extensively studied, owing to its low cost, good volumetric hydrogen density, and easy tailoring of hydrogenation thermodynamics by elemental substitution. All these positive aspects make this material promising for large-scale applications of solid-state hydrogen storage. On the other hand, activation and kinetic issues should be amended and the role of elemental substitution should be further understood. This work investigates the thermodynamic changes induced by the variation of Ti content along the homogeneity range of the TiFe phase (Ti:Fe ratio from 1:1 to 1:0.9) and of the substitution of Mn for Fe between 0 and 5 at.%. In all considered alloys, the major phase is TiFe-type together with minor amounts of TiFe2 or \b{eta}-Ti-type and Ti4Fe2O-type at the Ti-poor and rich side of the TiFe phase domain, respectively. Thermodynamic data agree with the available literature but offer here a comprehensive picture of hydrogenation properties over an extended Ti and Mn compositional range. Moreover, it is demonstrated that Ti-rich alloys display enhanced storage capacities, as long as a limited amount of \b{eta}-Ti is formed. Both Mn and Ti substitutions increase the cell parameter by possibly substituting Fe, lowering the plateau pressures and decreasing the hysteresis of the isotherms. A full picture of the dependence of hydrogen storage properties as a function of the composition will be discussed, together with some observed correlations.