This paper introduces a novel high-order, cell-centered Lagrangian method tailored for curvilinear meshes. To ensure stable curvilinear mesh motions and avoid the algorithmic complexity associated with special mesh stabilization techniques, this method exploits the subcell-wise computational degrees of freedom inherent in the well-established spectral volume (SV) method. It is demonstrated that the numerical results obtained from the SV reconstruction can serve as direct inputs for nodal solvers, yielding favorable mesh node velocities. For cells potentially containing discontinuities, the classical finite volume (FV) reconstruction is applied to each subcell within these cells instead. This subcell-wise reconstruction, analogous to common cell-wise limitations or reconstructions, can effectively reduce unphysical oscillations. More importantly, it is capable of suppressing spurious mesh motions following the same principle of not introducing stabilization techniques such as input correction. The proposed Lagrangian method is completely designed in the computational space with transformed governing equations. Hence, both of the aforementioned SV and FV reconstructions can be performed with the same ease as on uniform and time-independent Eulerian meshes and achieve high-order accuracy. The method further benefits from simplicity and uniformity in its implementation, the compactness of the reconstruction formulae utilized, and the efficiency of the iteration-free nodal solver employed. It exhibits consistently satisfying performance across various numerical tests, including those involving significant mesh deformation and extreme conditions such as low-pressure states accompanied by strong shocks.