Context. The origin of protostellar jets remains a major open question in star formation. Magnetohydrodynamical (MHD) disc winds are an important mechanism to consider, because they would have a significant impact on planet formation and migration. Aims. We wish to test the origins proposed for the extended hot H-2 at 2000 K around the atomic jet from the T Tauri star DG Tau, in order to constrain the wide-angle wind structure and the possible presence of an MHD disc wind in this prototypical source. Methods. We present spectro-imaging observations of the DG Tau jet in H-2 1-0 S(1) with 0.'' 12 angular resolution, obtained with SINFONI/VLT. Thanks to spatial deconvolution by the point spread function and to careful correction for wavelength calibration and for uneven slit illumination (to within a few km s(-1)), we performed a thorough analysis and modeled the morphology and kinematics. We also compared our results with studies in [Fe II], [O I], and FUV-pumped H-2, Absolute flux calibration yields the H2 column/volume density and emission surface, and narrows down possible shock conditions. Results. The limb-brightened H-2 1-0 S(1) emission in the blue lobe is strikingly similar to FUV-pumped H-2 imaged 6 yr later, confirming that they trace the same hot gas and setting an upper limit = 4 yr but undetected in [Fe II] by SINFONI, probably due to strong iron depletion. The rim thickness <= 14 AU rules out excitation by C-type shocks, and J-type shock speeds are constrained to similar or equal to 10 km s(-1). Conclusions. We find that explaining the H-2 1-0 S(1) wide-angle emission with a shocked layer requires either a recent outburst (15 yr) into a pre-existing ambient outflow or an excessive wind mass flux. A slow photoevaporative wind from the dense irradiated disc surface and an MHD disc wind heated by ambipolar diffusion seem to be more promising and need to be modeled in more detail. Better observational constraints on proper motion and rim thickness would also be crucial for clarifying the origin of this structure.