  Standard analyses of the reionization history of the universe from Planck
cosmic microwave background (CMB) polarization measurements consider only the
overall optical depth to electron scattering ($\tau$), and further assume a
step-like reionization history. However, the polarization data contain
information beyond the overall optical depth, and the assumption of a step-like
function may miss high redshift contributions to the optical depth and lead to
biased $\tau$ constraints. Accounting for its full reionization information
content, we reconsider the interpretation of Planck 2015 Low Frequency
Instrument (LFI) polarization data using simple, yet physically-motivated
reionization models. We show that these measurements still, in fact, allow a
non-negligible contribution from metal-free (Pop-III) stars forming in
mini-halos of mass $M \sim 10^5-10^6 M_\odot$ at $z \gtrsim 15$, provided this
mode of star formation is fairly inefficient. Our best fit model includes an
early, self-regulated phase of Pop-III star formation in which the reionization
history has a gradual, plateau feature. In this model, $\sim$20\% of the volume
of the universe is ionized by $z \sim 20$, yet it nevertheless provides a good
match to the Planck LFI measurements. Although preferred when the full
information content of the data is incorporated, this model would spuriously be
disfavored in the standard analysis. This preference is driven mostly by excess
power from E-mode polarization at multipoles of $10 \lesssim \ell \lesssim 20$,
which may reflect remaining systematic errors in the data, a statistical
fluctuation, or signatures of the first stars. Measurements from the Planck
High Frequency Instrument (HFI) should be able to confirm or refute this hint
and future cosmic-variance limited E-mode polarization surveys can provide
substantially more information on these signatures
