A new finite-volume flow solver based on the hybrid Cartesian immersed boundary (IB) framework is developed for the solution of high-speed inviscid compressible flows. The IB method adopts a sharp-interface approach, wherein the boundary conditions are enforced on the body geometry itself. A key component of the present solver is a novel reconstruction approach, in conjunction with inverse distance weighting, to compute the solutions in the vicinity of the solid-fluid interface. We show that proposed reconstruction leads to second-order spatial accuracy while also ensuring that the discrete conservation errors diminish linearly with grid refinement. Investigations of supersonic and hypersonic inviscid flows over different geometries are carried out for an extensive validation of the proposed flow solver. Studies on cylinder lift-off and shape optimisation in supersonic flows further demonstrate the efficacy of the flow solver for computations with moving and shape-changing geometries. These studies conclusively highlight the capability of the proposed IB methodology as a promising alternative for robust and accurate computations of compressible fluid flows on nonconformal Cartesian meshes. Copyright © 2017 John Wiley & Sons, Ltd.