When loading experiments are repeated on different samples, qualitatively different results can occur. This is due to factors such as geometric imperfections, load asymmetries, unevenly stressed regions or uneven material distributions created by manufacturing processes. This fact makes designing robust thin-walled structures difficult. One numerical strategy for exploring these different possible responses is to impose various initial imperfections on the geometry before loading, leading to different final solutions. However, this strategy is tedious, error-prone and gives an incomplete picture of the possible buckled configurations of the system. The present study demonstrates how a deflation strategy can be applied to obtain multiple solutions for the more robust design of thin-walled structures under displacement controlled uniaxial compression. We first demonstrate that distinct initial imperfections trigger different sequences of instability events in the Saint Venant–Kirchhoff hyperelastic model. We then employ deflation to investigate multiple bifurcation paths without the use of initial imperfections. A key advantage of this approach is that it can capture disconnected branches that cannot easily be discovered by conventional arc-length continuation and branch switching algorithms. Numerical experiments are given for three types of aircraft stiffener profiles. Our proposed technique is shown to be a powerful tool for exploring multiple disconnected bifurcation paths without requiring detailed insight for designing initial imperfections. We hypothesise that this technique will be very useful in the design process of robust thin-walled structures that must consider a variety of bifurcation paths.