Vibration energy harvesters have been proposed as an autonomous power source for meeting the limited power requirements of present-day sensors and electronics that find extensive usage in structural health monitoring systems. Recent research reveals that nonlinear energy harvesters outperform their linear counterparts, designed to operate on the principle of resonance, owing to their wide frequency bandwidth which allows for better performance in realistic operational environments. Particularly, bi-stable energy harvesters designed to exploit piezoelectricity to achieve the mechanical to electrical energy conversion have been widely investigated in literature. Additionally, several investigations have been also proposed to enhance power conversion in linear harvesters by introducing nonlinear circuits, e.g. based on synchronized switching (SS). In this respect, unveiling the effects on the bandwidth and coexisting solutions in the response of strongly nonlinear electrical SS shunts interacting with multi-stable structures requires further investigation. In particular, synchronized switch harvesting on inductor (SSHI) circuits, when connected in parallel with bi-stable energy harvesters, facilitate an increase in the harvested voltage, thus allowing for higher power generation as compared to a standard shunt load. This paper investigates the effects of utilizing a SSHI circuit for enhancing the power output in the different dynamical regimes of a bi-stable energy harvester. In particular, we present a comprehensive study of the efficiency of the SSHI circuit when the response configuration of the system is shifted between coexisting dynamical states of the bi-stable harvester. The herein presented semi-analytical and numerical results show that the qualitative performance of the SSHI circuit is quite robust in terms of superiority over the standard load circuit. However, the quantitative nature of the increase in power harvested by the SSHI circuit is sensitive to the optimal load in the circuit, which in turn varies with the dynamical state of the system.