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.