  Mirror symmetry, a three dimensional $\mathcal{N}=4$ IR duality, has been
studied in detail for quiver gauge theories of the $ADE$-type (as well as their
affine versions) with unitary gauge groups. The $A$-type quivers (also known as
linear quivers) and the associated mirror dualities have a particularly simple
realization in terms of a Type IIB system of D3-D5-NS5-branes. In this paper,
we present a systematic field theory prescription for constructing 3d mirror
pairs beyond the $ADE$ quiver gauge theories, starting from a dual pair of
$A$-type quivers with unitary gauge groups. The construction involves a certain
generalization of the $S$ and the $T$ operations, which arise in the context of
the $SL(2,\mathbb{Z})$ action on a 3d CFT with a $U(1)$ 0-form global symmetry.
We implement this construction in terms of two supersymmetric observables --
the round sphere partition function and the superconformal index on $S^2 \times
S^1$. We discuss explicit examples of various (non-$ADE$) infinite families of
mirror pairs that can be obtained in this fashion. In addition, we use the
above construction to conjecture explicit 3d $\mathcal{N}=4$ Lagrangians for 3d
SCFTs, which arise in the deep IR limit of certain Argyres-Douglas theories
compactified on a circle.
