Promising the seamless integration of multiple transportation modes, Mobility-as-a-Service (MaaS) has gained popularity over the years, yet its effectiveness in enticing private car users and improving travel efficiency remains uncertain. This study explores the competitiveness of MaaS-enabled multi-modal travel options versus private car usage through equilibrium analysis. In addition to pricing that affects the fixed cost for travel, we examine the often overlooked inconvenience cost associated with multi-modal trips. We establish our analysis for the commuting problem of a one-origin-one-destination network, where a highway and a mass transit line connect the residential area and the central business district (CBD) area. Travelers choose their departure time and travel mode among auto, Park-and-Ride (PnR), ride-hailing and transit (RnT) to minimize their total travel cost. Inconvenience costs associated with searching for parking and waiting for ride-hailing are explicitly modeled. We analytically provide the mode share and departure time windows under all possible equilibria. Our findings reveal the complex nature of mode choice, distinctly affected by fixed and inconvenience costs, with demand playing an even more significant role. Notably, fixed costs set an entry fee to adopt different modes, and the feature of inconvenience costs affects the utilization of available transportation resources. More importantly, to benefit the overall system, we encourage maintaining a balanced mode share by implementing pricing and capacity strategies rather than aiming for a completer transition of private car users to MaaS.