Consider the asymmetric nearest-neighbor exclusion process (ASEP) on ${\mathbb Z}$ with single particle drift $γ>0$, starting from a Bernoulli product invariant measure $ν_ρ$ with density $ρ$. It is known that the position $X_{N}$ of a tagged particle, say initially at the origin, at time $N$ satisfies an a.s. law of large numbers $\frac{1}{N}X_N \rightarrow γ(1-ρ)$ as $N\uparrow\infty$. In this context, we study the `typical' behavior of the tagged particle and `bulk' density evolution subject to `atypical' events $\{X_N\geq AN\}$ or $\{X_N\leq AN\}$ for $A\neq γ(1-ρ)$. We detail different structures, depending on whether $A<0$, $0\leq A< γ(1-ρ)$, $γ(1-ρ)<A< γ$, or $A\geq γ$, under which these atypical events are achieved, and compute associated large deviation costs. Among our results is an `upper tail' large deviation principle in scale $N$ for $\frac{1}{N}X_N$.