Let
T
N
T_N
denote an
N
×
N
N\times N
Toeplitz matrix with finite,
N
N
independent symbol
a
\mathbfit {a}
. For
E
N
E_N
a noise matrix satisfying mild assumptions (ensuring, in particular, that
N
−
1
/
2
‖
E
N
‖
H
S
→
N
→
∞
0
{N^{-1/2}\|E_N\|_{{\mathrm {HS}}}}\to _{N\to \infty } 0
at a polynomial rate), we prove that the empirical measure of eigenvalues of
T
N
+
E
N
T_N+E_N
converges to the law of
a
(
U
)
\mathbfit {a}(U)
, where
U
U
is uniformly distributed on the unit circle in the complex plane. This extends results from [Forum Math. Sigma 7 (2019)] to the non-triangular setup and non-complex Gaussian noise, and confirms predictions obtained in [Linear Algebra Appl. 162/164 (1992), pp. 153–185] using the notion of pseudospectrum.