A small polygon is a polygon of unit diameter. The maximal perimeter of a convex equilateral small polygon with $n=2^s$ vertices is not known when $s \ge 4$. In this paper, we construct a family of convex equilateral small $n$-gons, $n=2^s$ and $s\ge 4$, and show that their perimeters are within $\pi^4/n^4 + O(1/n^5)$ of the maximal perimeter and exceed the previously best known values from the literature. For the specific cases where $n=32$ and $n=64$, we present solutions whose perimeters are even larger, as they are within $1.1 \times 10^{-5}$ and $2.1 \times 10^{-6}$ of the optimal value, respectively.