A new algorithm is proposed for the construction of Brezing-Weng-like elliptic curves such that polynomials defining the CM discriminant are linear. Using this construction, new families of curves with variable discriminants and embedding degrees of $k{in}{8,16,20,24}$, which were not covered by Freeman, Scott, and Teske [9], are presented. Our result is useful for constructing elliptic curves with larger and more flexible discriminants.