Increased reliance on renewables brings challenges to maintaining the electric networks' inertia and damping and poses concerns about network stabilization. The virtual synchronous generator (VSG) technique has been adopted for dealing with stability issues in networks with high Inverter-Based Resource (IBR) penetration. This study investigates how the VSG control interacts with the synchronous generator (SG) in interconnected networks, revealing the existence of a new type of operating mode (named as electromechanical-like mode) in such networks. Employing a state-space model, the interplay between VSG inverters and synchronous generators is examined and validated in a four-machine-two-area system. The influence of the VSG's virtual inertia constant on oscillation damping is also evaluated. Based on these evaluation outcomes, a virtual inertia controller is developed in the VSG model, referred to as VSG-H. A centralized control by coordinating multiple VSGs with the proposed virtual inertia control (named as VSG-HCC) is subsequently developed to improve the low-frequency oscillation mode damping. The gain values of the VSG-H controller are computed through Particle Swarm Optimization (PSO). The adaptability of the proposed centralized controller is validated across different connection locations and for multiple VSG penetration scenarios. The robustness of the proposed controller with multiple VSG inverters is also demonstrated for attaining small-signal stability in two-area-four-machine and 39-bus test power systems.