This article analyses the stability of thin-shell wormholes constructed from non-asymptotically flat wormholes and the vacuum Schwarzschild solution. The construction of these spherically symmetric thin shells focuses on a specific class of wormholes characterized by a shape function that is linearly dependent on the radial coordinate. This introduces angular defects, which can be either deficits or excesses in the solid angle. To analyze the stability of these structures, we employ linear perturbations around a static solution, using a master equation to describe the behavior of stable equilibrium regions. The study is systematically divided to examine both positive and negative surface energy densities, and it delves into various gravitational redshift functions. Finally, it is concluded that the interaction of an external force on the thin shell significantly influences the behaviour of stable regions. It is demonstrated that, in certain cases, the matter supporting the thin shell may be non-exotic, fully satisfying all energy conditions.