Is there something similar to the small object argument, but related to a *chain* of factorization systems on a category $\cal C$?

It is easy to see that one can give a chain of "generating morphisms" $J_1\subseteq \cdots \subseteq J_n$ and obtain a reversed chain passing to the orthogonals $$ J_1^\perp\supseteq \cdots \supseteq J_n^\perp $$ If each $J_\alpha$ is a set of morphisms with small co/domains this, using the SOA one time for each $J_\alpha$, entails that there is a chain of factorization systems $$ \big({}^\perp(J_n^\perp),J_n^\perp \big) \le \cdots \le \big({}^\perp(J_1^\perp),J_1^\perp \big) $$

Series of Bousfield localizations of a given model category are a natural example of this. Is there some reference where this thing is defined from the purely categorical POV (and where, for example, the definition is stated for a *transfinite* chain $\alpha\to FS(\cal C)$)?