  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 disc spectrum (seen most
prominently in 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 entered 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
disc or the propelled matter. During the reflaring state, a few episodes with
stronger LP level (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, similar to AGN jets.
