Tunable diode laser absorption spectroscopy tomography (TDLAST) has been widely applied for imaging two-dimensional distributions of industrial flow-field parameters, e.g., temperature and species concentration. Two main interested imaging objectives in TDLAST are the local combustion and its radiation in the entire sensing region. State-of-the-art algorithms were developed to retrieve either of the two objectives. In this paper, we address the both by developing a novel multi-output imaging neural network, named as Spatially Progressive Neural Network (SpaProNet). This network consists of locally and globally prioritized reconstruction stages. The former enables hierarchical imaging of the finely resolved and highly accurate local combustion, but coarsely resolved background. The later retrieves a fine-resolved image for the entire sensing region, at the cost of slightly trading off the reconstruction accuracy in the combustion zone. Furthermore, the proposed network is driven by the hydrodynamics of the real reactive flows, in which the training dataset is obtained from large eddy simulation. The proposed SpaProNet is validated by both simulation and lab-scale experiment. In all test cases, the visual and quantitative metric comparisons show that the proposed SpaProNet outperforms the existing methods from the following two perspectives: a) the locally prioritized stage provides ever-better accuracy in the combustion zone; b) the globally prioritized stage shows turbulence-indicative accuracy in the entire sensing region for diagnosis of heat radiation from the flame and flame-air interactions.