Universal hitting sets are sets of words that are unavoidable: every long enough sequence is hit by the set (i.e., it contains a word from the set). There is a tight relationship between universal hitting sets and minimizers schemes, where minimizers schemes with low density (i.e., efficient schemes) correspond to universal hitting sets of small size. Local schemes are a generalization of minimizers schemes which can be used as replacement for minimizers scheme with the possibility of being much more efficient. We establish the link between efficient local schemes and the minimum length of a string that must be hit by a universal hitting set. We give bounds for the remaining path length of the Mykkeltveit universal hitting set. Additionally, we create a local scheme with the lowest known density that is only a log factor away from the theoretical lower bound.