We derive and analyze the confinement potential of the Cornell type within the framework of the generalized soft wall holographic model that includes a parameter controlling the intercept of the linear Regge spectrum. In the phenomenology of Regge trajectories, this parameter is very important for the quantitative description of experimental data. Our analysis shows that the ‘linear plus Coulomb’ confinement potential obtained in the scalar channel is quantitatively consistent with the phenomenology and lattice simulations while the agreement in the vector channel is qualitative only. This result indicates the key role of the vacuum scalar sector in the formation of the confinement potential. As a by-product the overall consistency of our holographic description of confinement potential seems to confirm the glueball nature of the scalar meson f ₀(1500).