In this paper, we characterize random eigenspaces with a non-intrusive method based on the decoupling of random eigenvalues from their corresponding random eigenvectors. This method allows us to estimate the first statistical moments of the random eigenvalues of the system with a reduced number of deterministic finite element computations. The originality of this work is to adapt the method used to estimate each random eigenvalue depending on a global accuracy requirement. This allows us to ensure a minimal computational cost. The stochastic model of the structure is thus reduced by exploiting specific properties of random eigenvectors associated with the random eigenfrequencies being sought. An indicator with no additional computation cost is proposed to identify when the method needs to be enhanced. Finally, a simple three-beam frame and an industrial structure illustrate the proposed approach.