In many applications, the detection of changing gaseous or liquid concentration within an environment is accomplished by monitoring the shift in resonance frequency of a microelectromechanical system designed to adsorb the target analyte. Recently, mass sensing using the onset and crossing of a dynamic bifurcation has been shown to reduce the mass threshold which may be detected. This approach effectively replaces detection of an analog quantity resolved by hardware capability (phase shift or resonant frequency) with a digital quantity having fundamental resolution restricted by system noise (crossing the bifurcation). While promising, successful sensing with oscillators continually excited near a system bifurcation is practically limited in performance by repeatable characteristics close to the critical crossing frequency and the passive detection ability of the sensors has not yet extended to mass quantization over a period of time. In this research, we explore an alternative method to exploit bifurcation for mass sensing by utilizing a new sensor system composed of a small bistable element within a primary linear host sensor that helps alleviate these concerns. The proposed system design provides adjustable control of the rate at which the bifurcation is crossed, helping to tailor the sensitivities of the system encountered in the transition region, introduces new bifurcations to exploit, and lends the opportunity to utilize the numerous bifurcation phenomena sequentially to denote mass accumulation quantity occurring between consecutive jump events. The conceptual underpinnings of the method are presented in detail and example operational trials are demonstrated by simulation to expound its operation and adjustability. Discussion is provided to evaluate the system in terms of existing bifurcation-based mass sensing approaches and to outline remaining goals.