Consider using a data structure that maintains elements in a way that allows efficient access to the smallest elements, such as a min-heap or a selection algorithm optimized for small k.

Explore the idea of partial sorting or using a modified version of quickselect that stops early once the kth smallest element is found, leveraging the fact that k is small.

Investigate the use of a tournament tree or a binary heap where you can extract the minimum element k times, but optimize the process to avoid the full O(n log n) time complexity by stopping after k extractions.