The phenomena of self-sustained shock-wave oscillations over conical bodies with a blunt axisymmetric base subject to uniform high-speed flow are investigated in a hypersonic wind tunnel at Mach number $M = 6$ . The flow and shock-wave dynamics is dictated by two non-dimensional geometric parameters presented by the three length scales of the body, two of which are associated with the conical forebody and one with the base. Time-resolved schlieren imagery from these experiments reveals the presence of two disparate states of shock-wave oscillations in the flow, and allows for the mapping of unsteadiness boundaries in the two-parameter space. Physical mechanisms are proposed to explain the oscillations and the transitions of the shock-wave system from steady to oscillatory states. In comparison with the canonical single-parameter problem of shock-wave oscillations over spiked-blunt bodies reported in literature, the two-parameter nature of the present problem introduces distinct elements to the flow dynamics.