Few months ago I have posted this question on MO, but I must admit that at the time, admittedly, I had no idea on how technical proof-theoretic considerations can be. I have decided to revise this question, and I thought that asking a new question is a better option that rewriting the old one (especially given that the other one has a partial answer to old question).
Given theory $T$ and recursive relation $\prec$, we say that $T$ proves $\prec$ to be well-ordered if $T$ proves that $\prec$ is a linear order and that $\forall X:((\forall n\prec m:n\in X)\Rightarrow n\in X)\Rightarrow\forall n:n\in X$.
We define proof-theoretic ordinal (PTO) of $T$ to be supremum of order types of relations which $T$ proves to be well-order.
Which recursive ordinals are PTOs of some theory extending $RCA_0$? How about theories extending $ACA_0$?
Side question: What is the PTO of theory $ACA_0+$ "$\varepsilon_0$ is well-ordered", where with $\varepsilon_0$ I mean a canonical well-ordering with order type $\varepsilon_0$?