Kick is a simple computational method that enables stochastic searches for stable structures of systems of up to similar to 15 atoms. We propose two new filters to be applied with Kick to enforce minimum bond lengths between all pair of atoms and connectivity of all atoms in candidate structures. As this Restricted Kick method rejects a rapidly increasing number of structures with increasing system size, we furthermore devised the restricted Random Assembly Stochastic Search method, which randomly assembles structures that automatically satisfy these filters. Significant improvements in convergence rate over the original Kick method are obtained for several well-established examples.