We present an hr-adaptivity framework for morphing a given mesh to fit a target surface prescribed as the zero isocontour of a discrete function. In this framework, high-order meshing is posed as a variational minimization problem that depends on the mesh quality prescribed via the target matrix optimization paradigm (TMOP) and position of a subset of mesh nodes with respect to the target surface. The proposed formulation ensures that the variational problem is converged and mesh quality degradation near the surface is limited, even when the mesh topology is incompatible with the target surface. Additionally, a mesh subset-based approach and h-refinement is introduced to efficiently increase fitting accuracy while reducing the computational cost of the mesh morphing problem. The hr-adaptivity technique extends to different element types in two- and three-dimensions, and can be used in existing finite element and spectral element frameworks to obtain high-order body-fitted meshes. Various numerical experiments demonstrate the robustness and accuracy of the fitting approach for problems of practical interest such as Lagrangian hydrodynamics and topology optimization.