We present two distinct, complementary methods for improving axial resolution in three-dimensional structured illumination microscopy (3D SIM) with minimal or no modification to the optical system. First, we show that placing a mirror directly opposite the sample enables 4-beam interference with higher spatial frequency content than 3D SIM illumination, offering near-isotropic imaging with ~120 nm lateral and 160 nm axial resolution. Second, we develop an improved deep learning method that can be directly applied to 3D SIM data, obviating the need for additional hardware. This procedure results in ~120 nm isotropic resolution and can be combined with denoising to facilitate volumetric imaging spanning dozens of time points. We demonstrate the potential of these advances by imaging a variety of cellular samples, delineating the nanoscale distribution of vimentin and microtubule filaments, observing the relative positions of caveolar coat proteins and lysosomal markers, and visualizing rich cytoskeletal dynamics within T-cells in the early stages of immune synapse formation.