  We study 3d theories containing $\mathcal{N}=3$ Chern-Simons vector
multiplets coupled to the $\mathrm{SU}(N)^3$ flavour symmetry of 3d $T_N$
theories with Chern-Simons level $k_1$, $k_2$ and $k_3$. It was formerly
pointed out that these theories flow to infrared SCFTs with enhanced
$\mathcal{N}=4$ supersymmetry when $1/k_1+1/k_2+1/k_3=0$. We examine
superconformal indices of these theories which reveal that supersymmetry of the
infrared SCFTs may get enhanced to $4 \leq \mathcal{N} \leq 6$ if such a
condition is satisfied. Moreover, even if the Chern-Simons levels do not obey
the aforementioned condition, we find that there is still an infinite family of
theories that flows to infrared SCFTs with $\mathcal{N}=4$ supersymmetry. The
't Hooft anomalies of the one-form symmetries of these theories are analysed.
As a by-product, we observe that there is generally a decoupled topological
sector in the infrared. When the infrared SCFTs have $\mathcal{N} \geq 4$
supersymmetry, we also study the Higgs and Coulomb branch limits of the indices
which provide geometric information of the moduli space of the theories in
question in terms of the Hilbert series.
