  Let $\{Z_n, n\geq 0\}$ be a supercritical branching process in an independent
and identically distributed random environment. We prove Cram\'{e}r moderate
deviations and Berry-Esseen bounds for $\ln (Z_{n+n_0}/Z_{n_0})$ % under the
annealed law, uniformly in $n_0 \in \mathbb{N}$, which extend the corresponding
results by Grama et al. (Stochastic Process.\ Appl. 2017) established for
$n_0=0$. The extension is interesting in theory, and is motivated by
applications. A new method is developed for the proofs; some conditions of
Grama et al. (2017) are relaxed in our present setting. An example of
application is given in constructing confidence intervals to estimate the
criticality parameter in terms of $\ln(Z_{n+n_0}/Z_{n_0})$ and $n$.
