The HTS insert for the 32 T superconducting magnet is comprised of two dry wound, double pancake REBCO coils of different lengths, nested within an LTS outsert. The high fields present within the HTS coils create large axial compression forces and significant radial expansion. The mechanical solutions implemented to account for the effects of axial and radial coil movement are described. These solutions are applied throughout many of the mechanical subsystems in and between the HTS coils, including electrical terminations, electrical bussing, external reinforcing of the windings, and suspension of the HTS coils within the outsert. Experiences from the assembly in preparation for the initial operation of the magnet are also shared.