Structural, magnetic and magnetocaloric properties of Mn₃Sn_1-xZn_xC antiperovskite carbides have been studied. With increasing Zn content the first-order magnetic transition (FOMT) is weakened. The Curie temperature (T_C) reduces first from 273 to 197 K and when x > 0.3, T_C increases, reaching its maximum of 430 K for x = 1.0. An increase in T_C is accompanied by pronounced changes in magnetic behaviour and a significant rise in magnetization from 21.82(4) to 76.2(2) Am²kg⁻¹ for x = 0.8 in the maximum applied magnetic field of 5 T. Neutron powder diffraction (NPD) was employed to study the magnetic structure of Mn₃Sn_1-xZn_xC compounds. The refinement of the NPD data for x = 0.3 revealed a magnetic structure with propagation vector k = (½,½,0) with a decrease in the canted antiferromagnetic (AFM) moment, which results in a reduction of the negative volume change at the magnetic transition and a decrease in the magnetocaloric effect (MCE). For x = 0.4, the magnetic structure is described by a propagation vector k = (½,½,½) for the AFM moment which dominates at low temperature, with the presence of a minor ferromagnetic (FM) component with a k = (0, 0, 0) propagation vector, which confirms the presence of the ferrimagnetic (FiM) state. For a higher Zn content (x = 0.6), the magnetic moment originates mainly from the FM component found on three independent Mn positions and an additional AFM moment oriented in the a-b plane. The results presented confirm the presence of competing AFM-FM interactions in Mn₃Sn_1-xZn_xC antiperovskite carbides.