A \({[}z,r;g{]}\) -mixed cage is a mixed graph of minimum order such that each vertex has z in-arcs, z out-arcs, r edges, and it has girth g, and the minimum order for \({[}z,r;g{]}\) -mixed graphs is denoted by n[z, r; g]. In this paper, we present an infinite family of mixed graphs with girth 6 that improves, in some cases, the families that we give in G. Araujo-Pardo and L. Mendoza-Cadena. On Mixed Cages of girth 6, arXiv:2401.14768v2. In particular, if q is an even prime power, we construct a family of graphs that satisfies \(n[\frac{q}{4},q;6]\le 4q^{2}-4\) , and if q is an odd prime power, and \(\frac{q-3}{2}\) is odd, then our family satisfies that \(n[\frac{q-1}{4},q;6]\le 4q^2-4\) , otherwise \(n[\frac{q-3}{4},q;6]\le 4q^2-4\) .
