Protostellar jets are tightly connected to the accretion process and regulate the angular momentum balance of accreting star-disk systems. The DG Tau jet is one of the best-studied protostellar jets and contains plasma with temperatures ranging over three orders of magnitude within the innermost 50AU of the jet. We present new Hubble Space Telescope (HST) far-ultraviolet (FUV) long-slit spectra spatially resolving the C IV emission (T similar to 10(5) K) from the jet for the first time, in addition to quasi-simultaneous HST observations of optical forbidden emission lines ({[}O I], {[}N II], {[}S II], and {[}O III]) and fluorescent H-2 lines. The C iv emission peaks at approximate to 42AU from the stellar position and has a FWHM of approximate to 52AU along the jet. Its deprojected velocity of around 200 km s(-1) decreases monotonically away from the driving source. In addition, we compare our HST data with the X-ray emission from the DG Tau jet. We investigate the requirements to explain the data by an initially hot jet compared to local heating. Both scenarios indicate a mass loss by the T similar to 10(5) K jet of similar to 10(-9) M-circle dot yr(-1), i.e., between the values for the lower temperature jet (T approximate to 10(4) K) and the hotter X- ray emitting part (T greater than or similar to 10(6) K). However, a simple initially hot wind requires a large launching region (similar to 1 AU), and we therefore favor local heating.