Let $Γ$ be an antipodal distance-regular graph with diameter $4$ and eigenvalues $θ_0>θ_1>θ_2>θ_3>θ_4$. Then $Γ$ is tight in the sense of Jurišić, Koolen, and Terwilliger [12] whenever $Γ$ is locally strongly regular with nontrivial eigenvalues $p:=θ_2$ and $-q:=θ_3$. Assume that $Γ$ is tight. Then the intersection numbers of $Γ$ are expressed in terms of $p$, $q$, and $r$, where $r$ is the size of the antipodal classes of $Γ$. We denote $Γ$ by $\mathrm{AT4}(p,q,r)$ and call this an antipodal tight graph of diameter $4$ with parameters $p,q,r$. In this paper, we give a new feasibility condition for the $\mathrm{AT4}(p,q,r)$ family. We determine a necessary and sufficient condition for the second subconstituent of $\mathrm{AT4}(p,q,2)$ to be an antipodal tight graph. Using this condition, we prove that there does not exist $\mathrm{AT4}(q^3-2q,q,2)$ for $q\equiv3$ $(\mathrm{mod}~4)$. We discuss the $\mathrm{AT4}(p,q,r)$ graphs with $r=(p+q^3)(p+q)^{-1}$.