  In the paper, we consider the extended Gross-Witten-Wadia unitary matrix
model by introducing a logarithmic term in the potential. The partition
function of the model can be expressed equivalently in terms of the Toeplitz
determinant with the $(i,j)$-entry being the modified Bessel functions of order
$i-j-\nu$, $\nu\in\mathbb{C}$. When the degree $n$ is finite, we show that the
Toeplitz determinant is described by the isomonodromy $\tau$-function of the
Painlev\'{e} III equation. As a double scaling limit, %In the double scaling
limit as the degree $n\to\infty$, we establish an asymptotic approximation of
the logarithmic derivative of the Toeplitz determinant, expressed in terms of
the Hastings-McLeod solution of the inhomogeneous Painlev\'{e} II equation with
parameter $\nu+\frac{1}{2}$. The asymptotics of the leading coefficient and
recurrence coefficient of the associated orthogonal polynomials are also
derived. We obtain the results by applying the Deift-Zhou nonlinear steepest
descent method to the Riemann-Hilbert problem for orthogonal polynomials on the
Hankel loop. The main concern here is the construction of a local parametrix at
the critical point $z=-1$, where the $\psi$-function of the Jimbo-Miwa Lax pair
for the inhomogeneous Painlev\'{e} II equation is involved.
