We continue the study of the lower central series of a free associative algebra, initiated by Feigin and Shoikhet (2007) [FS]. We generalize via Schur functors the constructions of the lower central series to any symmetric tensor category; specifically we compute the modified first quotient B1, and second and third quotients , and of the series for a free algebra T(V) in any symmetric tensor category, generalizing the main results of Feigin and Shoikhet (2007) [FS] and Arbesfeld and Jordan (2010) [AJ]. In the case Am|n:=T(Cm|n), we use these results to compute the explicit Hilbert series. Finally, we prove a result relating the lower central series to the corresponding filtration by two-sided associative ideals, confirming a conjecture from Etingof et al. (2009) [EKM], and another one from Arbesfeld and Jordan (2010) [AJ], as corollaries.