We report on an optical photometric and polarimetric campaign on the accreting millisecond X-ray pulsar (AMXP) SAX J1808.4-3658 during its 2019 outburst. The emergence of a low-frequency excess in the spectral energy distribution in the form of a red excess above the disk spectrum (seen most prominently in the z, i, and R bands) is observed as the outburst evolves. This is indicative of optically thin synchrotron emission due to a jet, as seen previously in this source and in other AMXPs during outburst. At the end of the outburst decay, the source enters a reflaring state. The low-frequency excess is still observed during the reflares. Our optical (BVRI) polarimetric campaign shows variable linear polarization (LP) throughout the outburst. We show that this is intrinsic to the source, with low-level but significant detections (0.2%-2%) in all bands. The LP spectrum is red during both the main outburst and the reflaring state, favoring a jet origin for this variable polarization over other interpretations, such as Thomson scattering with free electrons from the disk or the propelled matter. During the reflaring state, a few episodes with stronger LP levels (1%-2%) are observed. The low-level, variable LP is suggestive of strongly tangled magnetic fields near the base of the jet. These results clearly demonstrate how polarimetry is a powerful tool for probing the magnetic field structure in X-ray binary jets, as for active galactic nuclei jets.