  We consider a family of solutions to the Painlev\'e II equation $$
u''(x)=2u^3(x)+xu(x)-\alpha \qquad \textrm{with } \a \in \mathbb{R} \cut \{0\},
$$ which have infinitely many poles on $(-\infty, 0)$. Using Deift-Zhou
nonlinear steepest descent method for Riemann-Hilbert problems, we rigorously
derive their singular asymptotics as $x \to -\infty$. In the meantime, we
extend the existing asymptotic results when $x\to +\infty$ from $\a-\frac{1}{2}
\notin \mathbb{Z}$ to any real $\a$. The connection formulas are also obtained.
