  Data from the Belle Collaboration for associated production (with a light
unpolarized hadron) and single-inclusive production of transversely polarized
$\Lambda$-hyperons in $e^+e^-$ annihilation processes allowed to extract within
a TMD approach, for the first time, the $\Lambda$ polarizing fragmentation
function. Recent theoretical developments on the computation of cross sections
for single-inclusive hadron production in $e^+e^-$ annihilation processes,
within a Soft Collinear Effective Theory approach, combined with the CSS
formalism for the double-hadron production case, can be used to perform a
renewed analysis, adopting a full TMD factorization scheme. Preliminary results
and a comparison between the two analyses are presented.
