Meta-generalized gradient approximations (meta-GGAs) and local hybrid functionals generally depend on the kinetic energy density τ. For magnetic properties, this necessitates generalizations to ensure gauge invariance. In most implementations, τ is generalized by incorporating the external magnetic field. However, this introduces artifacts in the response of the density matrix and does not satisfy the iso-orbital constraint. Here, we extend previous approaches based on the current density to paramagnetic nuclear magnetic resonance (NMR) shieldings and electron paramagnetic resonance (EPR) g-tensors. The impact is assessed for main-group compounds and transition-metal complexes considering 25 density functional approximations. It is shown that the current density leads to substantial improvements—especially for the popular Minnesota and strongly constrained and appropriately normed (SCAN) functional families. Thus, we strongly recommend to use the current density generalized τ in paramagnetic NMR and EPR calculations with meta-GGAs.