Improved explicit Burgess-strength bounds for composite moduli

Consider the character sum $S(M,N)={\sum }_{n=M+1}^{M+N}\chi (n)$ for a non-principal Dirichlet character $\chi (n)$ to a (possibly composite) modulus $q$. Burgess proved that $S(M,N)\ll N^{1\mathrm{/}2}{q}^{3\mathrm{/}16+\epsilon }$, for any $\epsilon >0$. This was made explicit by Jain-Sharma et al., but only for moduli $q\ge \exp (\exp (9.594))\approx 8.03\times {10}^{6373}$. We prove a stronger result that holds when $q\ge 1.627\times {10}^{19}$.