Let {Zn, n ≥ 0} be a supercritical branching process in an independent and identically distributed random environment. We prove Cram´er moderate deviations and Berry-Esseen bounds for ln(Zn+n0/Zn0) uniformly in n0 ∈ N, which extend the corresponding results by Grama et al. (Stochastic Process. Appl. 2017) established for n0 = 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(Zn+n0/Zn0) and n