  Consider the transmission eigenvalue problem \[ (\Delta+k^2\mathbf{n}^2)
w=0,\ \ (\Delta+k^2)v=0\ \ \mbox{in}\ \ \Omega;\quad w=v,\ \ \partial_\nu
w=\partial_\nu v=0\ \ \mbox{on} \ \partial\Omega. \] It is shown in [12] that
there exists a sequence of eigenfunctions $(w_m, v_m)_{m\in\mathbb{N}}$
associated with $k_m\rightarrow \infty$ such that either
$\{w_m\}_{m\in\mathbb{N}}$ or $\{v_m\}_{m\in\mathbb{N}}$ are surface-localized,
depending on $\mathbf{n}>1$ or $0<\mathbf{n}<1$. In this paper, we discover a
new type of surface-localized transmission eigenmodes by constructing a
sequence of transmission eigenfunctions $(w_m, v_m)_{m\in\mathbb{N}}$
associated with $k_m\rightarrow \infty$ such that both
$\{w_m\}_{m\in\mathbb{N}}$ and $\{v_m\}_{m\in\mathbb{N}}$ are
surface-localized, no matter $\mathbf{n}>1$ or $0<\mathbf{n}<1$. Though our
study is confined within the radial geometry, the construction is subtle and
technical.
