Let $F_n(x)$ be the distribution function of a normalized sum of $n$ independent indentically distributed random variables. In the paper, estimations are obtained for the uniform deviation of the quantile $F_n^{-1}(y)$ from $\Phi^{-1}(y)$ in an interval $(a,b)$ with $a$ and $b$ tending to 0 and 1 respectively as fast as $\exp\{-kn^\alpha\}$, where $k>0$, $0<\alpha<1$. For summands bounded by a constant $c$, explicit formulas are given showing how constants in the estimations obtained depend on the parameter $c/\sigma$, where $\sigma^2$ is the variance of each summand.