Lattices and concept lattices are fundamental structures in mathematics and computer science, providing a robust framework for understanding order and hierarchy in various domains. The completion of a lattice, particularly through the Dedekind-MacNeille completion, is a well-established process that embeds a partially ordered set into a complete and minimal lattice. However, this process often results in the generation of a large and complex lattice, which may not be necessary or efficient for all applications. In this article, we introduce the concept of reduced subcontext completion, a novel approach that focuses on modifying an existing reduced subcontext rather than generating the entire sublattice. This method is analogous to the Dedekind-MacNeille completion but aims to maintain a simplified reduced subcontext while maintaining the join and meet operations, making it more practical for certain applications. We will first apply this approach to general lattice elements and then extend it to concept lattice elements, demonstrating its versatility and potential. Unlike traditional methods that seek to represent the complete sublattice corresponding to a known (possibly enlarged) context, our approach calculates a reduced subcontext representing the smallest sublattice encompassing the given elements or concepts. This distinction highlights the radical difference between the two methods and underscores the efficiency and practicality of the ReducedSubcontextCompletion algorithm, particularly in concept lattices.