Stress fibers (SFs), a contractile actin bundle in nonmuscle mesenchymal cells, are known to intrinsically sustain a constant level of tension or tensional stress, a process called cellular tensional homeostasis. Malfunction in this homeostatic process has been implicated in many diseases such atherosclerosis, but its mechanisms remain incompletely understood. Interestingly, the homeostatic stress in individual SFs is altered upon recruitment of α-smooth muscle actin in particular cellular contexts to reinforce the preexisting SFs. While this transition of the set-point stress is somewhat a universal process observed across different cell types, no clear explanation has been provided as to why cells end up possessing different stable stresses. To address the underlying physics, here we describe that imposing a realistic assumption on the nature of SFs yields the presence of multiple set-points of the homeostatic stress, which transition among them depending on the magnitude of the cellular tension. We analytically derive non-dimensional parameters that characterize the extent of the transition and predict that SFs tend to acquire secondary stable stresses if they are subject to as large a change in stiffness as possible or to as immediate a transition as possible upon increasing the tension. This is a minimal and simple explanation, but given the frequent emergence of force-dependent transformation of various subcellular structures in addition to that of SFs, the theoretical concept presented here would offer an essential guide to addressing potential common mechanisms governing complicated cellular mechanobiological responses.