The recently-invented tailored edge-ray concentrator (TERC) approach permits the design of compact two-stage high-flux solar collectors, with a focusing primary reflector and a non-imaging TERC secondary reflector. We present a new primary reflector shape based on the TERC approach and a secondary TERC tailored to its particular flux map, such that more compact concentrators emerge at flux concentration levels in excess of 90% of the thermodynamic limit. Calculations and raytrace simulation results are also offered which demonstrate that V-cone approximations to a wide variety of TERCs attain the concentration of the TERC to within a few percent. These V-cones represent practical secondary concentrators that may be superior to corresponding compound parabolic concentrator or trumpet secondaries.