The degree of curvature of fiber inclusions in fiber-reinforced composites impacts the elastic properties of the composite, but numerical studies of this effect are limited by the lack of efficient microstructure generators with fiber curvature control. We propose an extension to the fused Sequential Addition and Migration algorithm (Int. J. Numer. Methods Eng., 125(22), e7573, 2024) to allow for the direct control of the average fiber curvature. To this end, we prescribe an averaged curvature parameter for the entire microstructure, while also constraining local curvature for every fiber to a maximum value. While local curvature is unambiguously defined, the choice of the averaged curvature parameter is not obvious. We introduce as a novel curvature measure the deviation from straight curve (DSC), which computes as the trace of the local second-order fiber orientation tensor's covariance for a single fiber. Averaging this over all fibers leads to a scalar averaged curvature parameter. Using this parameter, we define an optimization procedure for generating curved-fiber reinforced microstructures. We compare the microstructure generation capabilities of the proposed extension with the prior algorithm without curvature control and find that the extended algorithm is capable of realizing a significantly wider spectrum of curvatures. Finally, we study the influence of fiber curvature on elastic properties, and find that for an example composite of glass-fiber reinforced polypropylene, increased fiber curvature leads to a reduction of the Young's modulus by as much as 10%.