The honeycomb cobaltate Na₂Co₂TeO₆ has recently been proposed as a candidate material for hosting field-induced quantum spin liquid (QSL) behavior. Here, we present a comprehensive thermodynamic study of its low-temperature, high-field phase diagram using magnetization, specific heat, and magnetocaloric-effect measurements down to 1 K. In zero field, we observe a weak residual moment that provides further insight into the nature of the magnetic ground state. For in-plane magnetic fields (B ∥ a^*), we identify three field-induced transitions at B_c1 ≃ 6 T, B_c2 ≃ 7.8 T, and B_c3 ≃ 10.4 T. The magnetic Grüneisen parameter and specific heat reveal clear thermodynamic signatures of these successive phase transitions enclosing two intermediate phases. Contrary to expectations for a field-induced QSL, the phase between B_c2 and B_c3 lacks enhanced magnetic entropy but instead shows behavior consistent with a distinct ordered state. Above B_c3, the absence of additional anomalies indicates a crossover to a conventional spin-polarized regime. Our results place stringent thermodynamic constraints on the proposed QSL scenario in Na₂Co₂TeO₆, calling for further microscopic investigations to establish the precise nature of the field-induced phases.