In this study, near-critical bifurcations of low Reynolds number (Re) flows past a pair of elliptic cylinders in the side-by-side arrangement are numerically investigated, and onsets of several distinct transition scenarios are addressed. A nested Cartesian-grid formulation, in combination with an effective immersed boundary method and a two-step fractional-step procedure, has been adopted to simulate the flows. The transition scenarios associated with various periodic, quasi-periodic, and biased flows, their bifurcation characteristics, corresponding critical Reynolds numbers, and phase-portraits are exploited to better understand the governing physics. From the global point of view, there appear variety of flow patterns within the investigated parameter space, 40 ⩽ Re ⩽ 300, 0.2 ⩽ G ⩽ 3.0 (G being the gap-ratio of the cylinders), and 1.5 ⩽ A ⩽ 3 (A is the cylinder aspect-ratio), which include, symmetric vortex shedding mode, semi-single/twin vortex street formations, asymmetric/deflected flows, stationary/biased flip-flopped-type vortex shedding, weakly-chaotic flows, and in-phase/anti-phase vortex synchronizations. We numerically present these flows by tuning Re quasi-stationary, and provide a broader understanding of the entire transition process. A comprehensive analysis of effects of Reynolds number, the gap-ratio, and the angle of incidence on different flow-induced forces on the cylinders is included in this regard. On the other hand, our simulated wakes with various non-zero incidence-angles are found to reveal a rich variety of instability induced weakly synchronized physical evolution characteristics, which remained virtually unexplored.