Aligned extracellular matrix fibers enable fibroblasts to undergo myofibroblastic activation and achieve elongated shapes. Activated fibroblasts are able to contract, perpetuating the alignment of these fibers. This poorly understood feedback process is critical in chronic fibrosis conditions, including cancer. Here, using fiber networks that serve as force sensors, we identify "3D perpendicular lateral protrusions" (3D-PLPs) that evolve from lateral cell extensions named twines. Twines originate from stratification of cyclic-actin waves traversing the cell and swing freely in 3D to engage neighboring fibers. Once engaged, a lamellum forms and extends multiple secondary twines, which fill in to form a sheet-like PLP, in a force-entailing process that transitions focal adhesions to activated (i.e., pathological) 3D-adhesions. The specific morphology of PLPs enables cells to increase contractility and force on parallel fibers. Controlling geometry of extracellular networks confirms that anisotropic fibrous environments support 3D-PLP formation and function, suggesting an explanation for cancer-associated desmoplastic expansion.