  In this paper, we prove a lower bound for $\underset{\chi \neq
\chi_0}{\max}\bigg|\sum_{n\leq x} \chi(n)\bigg|$, when $x= \frac{q}{(\log
q)^B}$. This improves on a result of Granville and Soundararajan for large
character sums when the range of summation is wide. When $B$ goes to zero, our
lower bound recovers the expected maximal value of character sums for most
characters.
