We report magnetic-field and angular-dependent high-resolution specific-heat measurements of the organic superconductor β''-(BEDT-TTF)2SF5CH2CF2SO3. When the magnetic field is aligned precisely within the conducting BEDT-TTF layer, at low temperatures a clear upturn of the upper critical field beyond the Pauli limit of 9.73 T is observed, hinting at the emergence of a Fulde-Ferrell-Larkin-Ovchinnikov state. This upturn disappears when the field is oriented out of plane by more than ∼0.5 deg. For smaller out-of-plane angles, the specific-heat anomaly at T(c) sharpens and a second peaky phase transition appears within the superconducting state.