Galactic black hole binaries produce powerful outflows which emit over almost the entire electromagnetic spectrum. Here, we report the first detection with the Herschel observatory of a variable far-infrared source associated with the compact jets of the black hole transient GX 339-4 during the decay of its recent 2010-2011 outburst, after the transition to the hard state. We also outline the results of very sensitive radio observations conducted with the Australia Telescope Compact Array, along with a series of near-infrared, optical (OIR) and X-ray observations, allowing for the first time the re-ignition of the compact jets to be observed over a wide range of wavelengths. The compact jets first turn on at radio frequencies with an optically thin spectrum that later evolves to an optically thick synchrotron emission. An OIR reflare is observed about 10d after the onset of radio and hard X-ray emission, likely reflecting the necessary time to build up enough density, as well as to have acceleration (e.g. through shocks) along an extended region in the jets. The Herschel measurements are consistent with an extrapolation of the radio inverted power-law spectrum, but they highlight a more complex radio to OIR spectral energy distribution for the jets.