In this paper, we present a construction of locally recoverable codes (LRCs) with multiple recovery sets using algebraic curves with many rational points. By leveraging separable morphisms between smooth projective curves and expanding the class of curves previously considered, we significantly generalize and enhance the framework. Our approach corrects certain inaccuracies in the existing literature while extending results to a broader range of curves, thereby achieving better parameters and wider applicability. In addition, the constructions presented here result in LRCs with large availability.