In this article, we study a branching random walk evolving in a (macroscopically) time-inhomogeneous environment. We compute the first two orders of the asymptotic of the maximal displacement. The ballistic first order is given by the solution of an optimization problem, while the second term is of order $n^{1/3}$. This result partially answers a conjecture of Fang and Zeitouni. Additionally, we obtain an asymptotic for the so-called consistent maximal displacement of the process, which also is of order $n^{1/3}$.