When do we get $CON(ZF)$ in transfinite progressions of consistency statements? Given the work of Turing and Feferman all arithmetical truths can be isolated through a transfinite progression of theories like $T_0=PA$, $T_{\beta+1}=T_β \ plus \ CON(T_\beta)$ and $T\lambda=\cup T\mu(\mu\prec\lambda)$ - when $\lambda$ is a limit ordinal - through all the recursive ordinals. What is the smallest ordinal $\sigma$ such that $T_\sigma$ proves CON(ZF)? How do such ordinals for arithmetical consistency statements align with proof theoretical ordinals? 
Edit: My question does not ask for the proof theoretic ordinal of ZF. 
Update: Phillip Welch gives a very readable account of such things as I hint to in comments concerning Feferman's work in an answer to a question here:
Pi1-sentence independent of ZF, ZF+Con(ZF), ZF+Con(ZF)+Con(ZF+Con(ZF)), etc.?
Update 2: My question was badly prepared, as evidenced also by the previous update and the comments in discussion. Noah Schweber kindly suggested that I unaccept his reply until more is clarified concerning my question as related to the Feferman style process I had in mind, and which through a detour into Shoenfield's recursive omega rule (non-constructively) captures all arithmetical truths. I would be surprised if Turing like collapses down to $\omega+1$ could occur in Feferman style processes. 
 A: Note that the progression $T_\alpha$ really isn't defined for ordinals but rather ordinal notations. Once we realize this, there is a disappointing answer to your question: for any true $\Pi^0_1$ sentence $\varphi$ (of which a consistency statement is an example), there is a notation $n$ for $\omega+1$ such that $T_n$ proves $\varphi$.
See this answer by Francois Dorais for more details.
This phenomenon breaks the initial hope of assigning an interesting ordinal to a theory $S$ measuring the difficulty of proving $S$'s consistency via iterated consistency statements. However, we can fix things by working below some fixed notation for a "large enough" ordinal: e.g. the ordinal $\epsilon_0$ has, in addition to really stupid notations, very natural notations, and we can work below such a notation to develop the fast-growing hierarchy.
So if we fix a notation $n$, it may be that some notation $m<_\mathcal{O}n$ for a smaller ordinal satisfies "$T_m$ proves $Con(ZF)$"; and if $n$ is "nice", this $m$ might be really interesting! Unfortunately this is putting the cart before the horse: in order to find such an $n$, we basically already need to know everything relevant about $ZFC$, including (at least something close to) its proof-theoretic ordinal.
A: There is no known explicit combinatorial description of the proof-theoretic ordinal of ZFC. Even much weaker set theories have so far defied explicit description.  For a recent account that gives some sense of the state of the art, see "Notes on some second-order systems of iterated inductive definitions and $\Pi_1^1$-comprehensions and relevant subsystems of set theory," by Kentaro Fujimoto, Annals of Pure and Applied Logic, 166 (2015), 409–463.
