An appropriate description of the state of matter that appears as a second order phase transition is tuned toward zero temperature, viz. quantum-critical point (QCP), poses fundamental and still not fully answered questions. Experiments are needed both to test basic conclusions and to guide further refinement of theoretical models. Here, charge and entropy transport properties as well as AC specific heat of the heavy-fermion compound CeRh0.58Ir0.42In5, measured as a function of pressure, reveal two qualitatively different QCPs in aﾠsinglematerial driven by a single non-symmetry-breaking tuning parameter. A discontinuous sign-change jump in thermopower suggests an unconventional QCP atﾠpc1ﾠaccompanied by an abrupt Fermi-surface reconstruction that is followed by a conventional spin-density-wave critical point atﾠpc2ﾠacross which the Fermi surface evolves smoothly to a heavy Fermi-liquid state. These experiments are consistent with some theoretical predictions, including the sequence of critical points and the temperature dependence of the thermopower in their vicinity.