By means of the incremental elastic modulus for small deformations superimposed on creep and subsequent recovery tests, we follow the structural state of various soft-jammed systems (yield stress fluids) in their solid regime. We demonstrate that the solid state of the material is associated with a persistent elastic network of constant elastic modulus up to yielding (solid-liquid transition), while progressively more additional elastoplastic elements are involved. The main features of these elastoplastic elements, i.e. increase of both the plastic and the elastic deformation components with the square of the shear stress, reveal the fundamental characteristics of a simple generic model (independent of material structure) describing the different components of the mechanical behavior of such systems in their solid regime.