This study investigated the effect of stress-induced martensite aging under tensile and compressive stresses on the functional and viscoelastic properties in Ni50.3Ti32.2Hf17.5 polycrystals containing dispersed H-phase particles up to 70 nm in size obtained by preliminary austenite aging at 873 K for 3 h. It was found that stress-induced martensite aging at 428 K for 12 h results in the appearance of a two-way shape memory effect of −0.5% in compression and +1.8% in tension. Moreover, a significant change in viscoelastic properties can be observed: an increase in internal friction (by 25%) and a change in elastic modulus in tensile samples. The increase in internal friction during martensitic transformation after stress-induced martensite aging is associated with the oriented growth of thermal-induced martensite. After stress-induced martensite aging, the elastic modulus of martensite (EM) increased by 8 GPa, and the elastic modulus of austenite (EA) decreased by 8 GPa. It was shown that stress-induced martensite aging strongly affects the functional and viscoelastic properties of material and can be used to control them.