We consider the Halfin-Whitt diffusion process $X_d(t)$, which is used, for example, as an approximation to the $m$-server $M/M/m$ queue. We use recently obtained integral representations for the transient density $p(x,t)$ of this diffusion process, and obtain various asymptotic results for the density. The asymptotic limit assumes that a drift parameter $β$ in the model is large, and the state variable $x$ and the initial condition $x_0$ (with $X_d(0)=x_0>0$) are also large. We obtain some alternate representations for the density, which involve sums and/or contour integrals, and expand these using a combination of the saddle point method, Laplace method and singularity analysis. The results give some insight into how steady state is achieved, and how if $x_0>0$ the probability mass migrates from $X_d(t)>0$ to the range $X_d(t)<0$, which is where it concentrates as $t\to\infty$, in the limit we consider. We also discuss an alternate approach to the asymptotics, based on geometrical optics and singular perturbation techniques.