In this work we study the partially constrained vielbein formulation of the
new quasidilaton theory of massive gravity which couples to both physical and
fiducial metrics simultaneously via a composite effective metric. This
formalism improves the new quasidilaton model since the Boulware-Deser ghost is
removed fully non-linearly at all scales. This also yields crucial implications
in the cosmological applications. We derive the governing cosmological
background evolution and study the stability of the attractor solution.