Self-assembly is a powerful approach for constructing colloidal crystals, where spheres, rods or faceted particles can build up a myriad of structures. Nevertheless, many complex or low-coordination architectures, such as diamond, pyrochlore and other sought-after lattices, have eluded self-assembly. Here we introduce a new design principle based on preassembled components of the desired superstructure and programmed nearest-neighbour DNA-mediated interactions, which allows the formation of otherwise unattainable structures. We demonstrate the approach using preassembled colloidal tetrahedra and spheres, obtaining a class of colloidal superstructures, including cubic and tetragonal colloidal crystals, with no known atomic analogues, as well as percolating low-coordination diamond and pyrochlore sublattices never assembled before.