  We perform a re-analysis of Belle data for the transverse $\Lambda$ and
$\bar\Lambda$ polarization in $e^+ e^-$ annihilation processes within a
transverse momentum dependent (TMD) factorization approach. We consider two
data sets, one referring to the associated production of $\Lambda$'s with a
light unpolarized hadron in an almost back-to-back configuration, and one for
the inclusive $\Lambda$ production, with the reconstruction of the thrust axis.
We adopt the Collins-Soper-Sterman framework and employ the recent formulation
on the factorization of single-inclusive hadron production. This extends a
previous phenomenological study carried out in a more simplified TMD approach,
leading to a new extraction of the polarizing fragmentation function (FF).
While confirming several features of the previous analysis, here we include the
proper QCD scale dependence of this TMD FF and carefully exploit the role of
its nonperturbative component. The compatibility, within a unique TMD
factorization scheme, of double and single-inclusive hadron production is
discussed in detail. Moreover, we consider another data set for inclusive
$\Lambda$ production, at much larger energies, from the OPAL Collaboration, in
order to test the consistency of the entire approach. Finally, we elaborate on
some fundamental issues like the role of $SU(2)$ isospin symmetry and the heavy
quark contributions.
