In this paper, we show new data structures maintaining approximate shortest paths in sparse directed graphs with polynomially bounded non-negative edge weights under edge insertions. We give more efficient incremental $(1+ε)$-approximate APSP data structures that work against an adaptive adversary: a deterministic one with $\tilde{O}(m^{3/2}n^{3/4})$ total update time and a randomized one with $\tilde{O}(m^{4/3}n^{5/6})$ total update time. For sparse graphs, these both improve polynomially upon the best-known bound against an adaptive adversary. To achieve that, building on the ideas of [Chechik-Zhang, SODA'21] and [Kyng-Meierhans-Probst Gutenberg, SODA'22], we show a near-optimal $(1+ε)$-approximate incremental SSSP data structure for a special case when all edge updates are adjacent to the source, that might be of independent interest. We also describe a very simple and near-optimal \emph{offline} incremental $(1+ε)$-approximate SSSP data structure. While online near-linear partially dynamic SSSP data structures have been elusive so far (except for dense instances), our result excludes using certain types of impossibility arguments to rule them out. Additionally, our offline solution leads to near-optimal and deterministic all-pairs bounded-leg shortest paths data structure for sparse graphs.