The $Δ$-Springer fibers $Y_{n,λ,s}$, introduced by Levinson, Woo, and the second author, generalize Springer fibers for $\mathrm{GL}_n(\mathbb{C})$ and give a geometric interpretation of the of the Delta Conjecture from algebraic combinatorics (at $t=0$). We prove that all irreducible components of the $Δ$-Springer fiber $Y_{n,n-1}=Y_{n,(1^{n-1}),n-1}$ are smooth. In fact, we prove that any intersection of irreducible components of $Y_{n,n-1}$ is a smooth Hessenberg variety which has the structure of an iterated Grassmannian fiber bundle. We then give a presentation of the singular cohomology ring of each irreducible component of $Y_{n,n-1}$ and a combinatorial formula for the Poincaré polynomial of an arbitrary union of intersections of irreducible components in terms of arm and leg statistics on Dyck paths.