Gravitational wave astronomy is expected to provide independent constraints on neutron star properties, such as their dense matter equation of state. This is possible with the measurements of binary components' tidal deformability, which alter the point-particle gravitational waveforms of the late inspiral phase of neutron-star binaries... Although current gravitational wave detectors are not sensitive enough for a precise determination of the individual tidal deformations of the components, a large number of combined observations with future detectors will decrease uncertainties in this quantity. Here we provide a first study of the tidal deformability effects due to the elasticity/solidity of the crust (hadronic phase) in a hybrid neutron star, as well as the influence of a quark-hadronic phase density jump on tidal deformations. We employ the framework of nonradial perturbations with zero frequency and study hadronic phases presenting elastic aspects when perturbed (with the shear modulus approximately $1\%$ of the pressure). We find that the relative tidal deformation change in a hybrid star with a perfect-fluid quark phase and a hadronic phase presenting an elastic part is never larger than about $2-4\%$ (with respect to a perfect-fluid counterpart). These maximum changes occur when the elastic region of a hybrid star is larger than approximately $60\%$ of the star's radius, which may happen when its quark phase is small and the density jump is large enough, or even when a hybrid star has an elastic mixed phase. For other cases, the relative tidal deformation changes due to an elastic crust are negligible ($10^{-5}-10^{-1}\%$), therefore unlikely to be measured even with third generation detectors. Thus, only when the size of the elastic hadronic region of a hybrid star is over half of its radius, the effects of elasticity could have a noticeable impact on tidal deformations. (read more)