  Based on an optimal rate wavelet series representation, we derive a local
modulus of continuity result with a refined almost sure upper bound for
fractional Brownian motion. \sloppy The obtained upper bound of the small
fractional Brownian increments is of order $\mathcal
O_{a.s.}\big(|h|^H\sqrt{\log\log |h|^{-1}}\big)$ as $|h|\to0$, and an upper
bound of its $p$th moment is provided, for any $p>0$. This result fills the gap
of the law of iterated logarithm for fractional Brownian motion, where the
moments' information of the random multiplier in the upper bound is missing.
With this enhanced upper bound and some new results on the distribution of the
maximum of fractional Brownian motion, we obtain a new and refined asymptotic
estimate of the upper-tail probability for a fractional Brownian motion to
first exit from a positive-valued barrier over time $T$, as $T\to+\infty$.
