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We investigate thermal Hall conductivity κxy of a J-K Kitaev-Heisenberg model with a Zeeman field in the (111) direction in the light of the recent debate surrounding the possible re-emergence of Ising topological order (ITO) and half-quantized κxy/T upon field-suppression of long-range magnetic order in Kitaev materials. We use the purification-based finite temperature Tensor Network approach making no prior assumptions about the nature of the excitations: Majorana, visons or spin waves. For purely Kitaev interactions and fields h/K & 0.02 sufficient to degrade ITO, the peak κxy/T monotonously decreases from half-quantization associated with lower fields a behavior reminiscent of vison fluctuation corrections. For higher fields h/K & 0.1, we find the results qualitatively consistent with a spin-wave treatment. In our J-K model (with ferro-K and antiferro-J), in the vicinity of field-suppressed magnetic order, we found κxy/T to be significant, with peak magnitudes exceeding half-quantization followed by a monotonous decrease with increasing h. We thus conclude that half-quantized thermal Hall effect in the vicinity of field suppressed magnetic order in our model, is a fine-tuning effect and is not associated with a Majorana Hall state with ITO.