Suppose there is a matrix $A$ that transforms vectors, $$ Y = A x $$ Now express this in some other coordinate system, with $x = B z, \,\, y = B w$, so \begin{align*} & Bw = A B z \\ \Rightarrow & w = B^{-1} A B z \end{align*} So $A$ expressed in the other system is $B^{-1} A B$.
What would be the equivalent in tensor notation, in particular of the $B^{-1}$? Here's what I'm trying \begin{align*} & y^i = A^i_j x_j \\ & \quad\quad x^j = B^j_k z^k \\ & \quad\quad y^i = B^i_m w^m \\ \text{so}\quad & B^i_m w^m = A^i_j B^j_k z_k \end{align*} Now what is the tensor equivalent of premultiplying by $B^{-1}$ on the left, in order to find what $A$ looks like in tensor notation in the new coordinate system?
The tensor equivalent is simply:
$$ (B^{-1})^i_j $$
defined to satisfy
$$ (B^{-1})^i_j B^j_k = \delta^i_k \qquad B^i_j (B^{-1})^j_k = \delta^i_k $$.
So you write
$$ w^m = (B^{-1})^m_i A^i_j B^j_k z^k $$
Though, I would really, really just advice you to use abstract index notation instead of the concrete tensor notation you seem to be using. The advantage is that you treat $z$ now as an element in your vector space $V$ and not as the coordinate representation, and $A$ the linear mapping from $V$ to itself instead of its matrix representation, so you don't have to worry about matrices coming from "changes of variables".