We report specific heat measurements of the heavy fermion superconductor CeCoIn5 in the vicinity of the superconducting critical field H(c2), with magnetic fields in the , , and  directions, and at temperatures down to 50 mK. The superconducting phase transition changes from second to first order for fields above 10 T for H parallel  and H parallel . In the same range of magnetic fields, we observe a second specific heat anomaly within the superconducting state. We interpret this anomaly as a signature of a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) inhomogeneous superconducting state. We obtain similar results for H parallel , with the FFLO state occupying a smaller part of the phase diagram.