Selberg sieve weights and sign changes of Kloosterman sums. II. Square-free moduli with at most four prime factors

Let $\text{Kl}(1;q)$ denote the normalized Kloosterman sum modulo $q$. We prove that, for each sign, there are $\gg X\mathrm{/}\log X$ square-free moduli $q\in (X,2X]$ with at most four prime factors for which $\text{Kl}(1;q)$ has that sign. The proof combines the Selberg sieve with equidistribution and mean-square estimates for Kloosterman sums; the uniform estimate for shifted sieve weights established in Part~I controls the contribution of moduli having small prime factors.