When an external magnetic field is applied parallel to the conducting layers in quasi-2D organic superconductors, a phase transition to a new, high-field, superconducting state with a spatially modulated superconducting order parameter can occur at the Clogston-Chandrasekhar paramagnetic limit H_p, as first predicted by Fulde, Ferrell, Larkin, and Ovchinnikov (FFLO). Candidate FFLO phases have been identified through NMR, RF penetration depth, magnetic-torque, and transport measurements in a number of organic superconductors; in the case of the archetypal 2D organic superconductor κ-(BEDT-TTF)₂Cu(NCS)_{2 ,} corroborating evidence has been provided by the calorimetric observation of a first order phase transition at H_p into a new high-field, higher-entropy superconducting phase. This evidence has been lacking, however, in other candidate FFLO systems. We report here low-temperature specific heat measurements in a second expected FFLO organic superconductor: λ-(BETS)₂GaCl₄ . For field-angles within 0.5° of plane-parallel, we observe an enhancement of the upper critical field superconducting phase boundary H_c2(T), a calorimetric signature of pancake vortex formation with decreasing field at H_p, and a field-induced phase transition between the upper and lower superconducting states.