A subset B of the ring \(\mathbb {Z}_n\) is referred to as a \(\ell \) -covering set if \(\{ ab \pmod n \mid 0\le a \le \ell , b\in B\} = \mathbb {Z}_n\) . We show that there exists a \(\ell \) -covering set of \(\mathbb {Z}_n\) of size \(O(\frac{n}{\ell }\log n)\) for all n and \(\ell \) , and how to construct such a set. We also provide examples where any \(\ell \) -covering set must have a size of \(\varOmega (\frac{n}{\ell }\frac{\log n}{\log \log n})\) . The proof employs a refined bound for the relative totient function obtained through sieve theory and the existence of a large divisor with a linear divisor sum. The result can be used to simplify a modular subset sum algorithm.