The role of the isovector rho-meson coupling on the symmetry energy in the relativistic mean-field theory has been investigated. In a multi-dimensional parameter search for an appropriate force within the framework of the non-linear σωϱ model the ϱ-meson coupling has been found to have a significant effect on the neutron skin of nuclei. It has been shown that a low value of ϱ-meson coupling strength remedies the defect of some of the existing parameter sets of providing a higher neutron skin thickness of nuclei. An extraordinarily good description of highly proton- as well as neutron-rich exotic nuclei has been found. It has been observed that the proper description of both proton-rich and neutron-rich nuclei is sensitive to the symmetry energy.