Every topologically transitive shift space, whether intrinsically ergodic or not, can be approximated from above by intrinsically ergodic systems.

Indeed, given a finite alphabet $A=\{1,2,\dots,p\}$ and a  closed $\sigma$-invariant set $X\subset A^\mathbb{Z}$ (everything works just the same for one-sided shifts), let $\mathcal{L}=\mathcal{L}(X) \subset A^* = \bigcup_{n\geq 1} A^n$ be the collection of all finite words that appear in some sequence $x\in X$.  Thus $X$ determines $\mathcal{L}$ and vice versa.  Let $\mathcal{F} = A^* \setminus \mathcal{L}$ be the set of forbidden words.  Now let $Y_n\subset A^\mathbb{Z}$ be the set of all sequences that do not contain any words in $\mathcal{F}$ of length $\leq n$.  Then $Y_n$ is a shift of finite type and $X= \bigcap_{n\geq 1} Y_n$.

Furthermore, $Y_n$ is topologically transitive and hence intrinsically ergodic by virtue of being an SFT.  To see this, choose any $v,w\in \mathcal{L}(Y_n)$ and write $v=v_1 v_2$, $w=w_1 w_2$ where $v_2$ and $w_1$ both have length exactly $n$.  Then by transitivity of $X$ there exists a word $u$ such that $v_2 u w_1 \in \mathcal{L}(X)$, and by the definition of $Y_n$ we have $v u w\in \mathcal{L}(Y_n)$, which shows that $Y_n$ is transitive.

Thus the counterexample given in the answer to your earlier question works here as well.  Actually, I'll point out that there are quite a broad class of such counterexamples, which can be constructed by looking at coded systems: these are shift spaces defined either in terms of a countable collection of generating words that are allowed to be freely concatenated, or equivalently in terms of a directed graph on countably many vertices with edges labeled from a finite alphabet.  There are plenty of examples of coded systems that are transitive but not intrinsically ergodic (see [this question][1] or [this answer][2], for example), and you can approximate coded systems from within by SFTs (or at least sofic shifts) in a very natural way:  just truncate the collection of generators to a finite set, or truncate the graph to a finite subgraph.

In another direction, I believe there is a paper of Gurevich in which certain quantitative conditions are given on the *rate* of approximation from outside by intrinsically ergodic systems that turn out to be sufficient to guarantee intrinsic ergodicity of $X$.  But I don't have the reference handy at the moment, and I'll have to wait until I'm in my office next week to dig up the paper and see if it's in fact relevant.

  [1]: http://mathoverflow.net/questions/26094/a-topologically-mixing-subshift-with-multiple-measures-of-maximal-entropy
  [2]: http://mathoverflow.net/questions/43564/transitive-shifts-with-multiple-fully-supported-mmes/94015#94015