The technique of floating shock fitting is adapted to the computation of the inviscid flowfield about circular cones in a supersonic freestream at angles of attack that exceed the cone half-angle. In those regions in which the governing conical equations are mixed elliptic-hyperbolic, the fully hyperbolic form is obtained by the addition of the temporal derivative. For cones of infinite extent, the flow maintains its conical nature up to the detachment point. Thus, the resulting equations are applicable over the complete range of freestream Mach numbers, angles of attack, and cone half-angles for which the bow shock is attached. An explicit finite-difference algorithm is used to obtain the solution by an unsteady relaxation approach. The bow shock, embedded crossflow shock, and vortical singularity in the leeward symmetry plane are all treated as floating discontinuities in a fixed computational mesh. The method yields excellent results for the bow and embedded shocks; however, the solution in the leeward symmetry plane exhibits viscous-Iike effects and does not appear to adequately predict the behavior of the vortical singularity.