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The two-temperature relativistic electron spectrum from a low-density (3×10^{17}  cm^{-3}) self-modulated laser wakefield accelerator (SM-LWFA) is observed to transition between temperatures of 19±0.65 and 46±2.45  MeV at an electron energy of about 100 MeV. When the electrons are dispersed orthogonally to the laser polarization, their spectrum above 60 MeV shows a forking structure characteristic of direct laser acceleration (DLA). Both the two-temperature distribution and the forking structure are reproduced in a quasi-3D osiris simulation of the interaction of the 1-ps, moderate-amplitude (a_{0}=2.7) laser pulse with the low-density plasma. Particle tracking shows that while the SM-LWFA mechanism dominates below 40 MeV, the highest-energy (>60  MeV) electrons gain most of their energy through DLA. By separating the simulated electric fields into modes, the DLA-dominated electrons are shown to lose significant energy to the longitudinal laser field from the tight focusing geometry, resulting in a more accurate measure of net DLA energy gain than previously possible.