We study the behavior of West's stack-sorting map $s$ on permutations whose last entry is also their least. Let $S_{n}':=\{\pi0\mid π\in S_n\}$ where $\pi0$ denotes the concatenation of $π$ and $0$. For each permutation $π\in S_n'$, we introduce a new combinatorial object known as the stack-sorting tableau $T_π$, which ultimately serves as the key ingredient in the first polynomial time algorithm for counting the number of $t$-stack-sortable permutations in $S_n'$. We then establish a precise relationship between the behavior of $s$ on $S_{n}'$ and on $S_{n}$.