Summary: The numerical solution of contact problems via the penalty method yields approximate satisfaction of contact constraints. The solution can be improved using augmentation schemes. However, their efficiency is strongly dependent on the value of the penalty parameter and usually results in a poor rate of convergence to the exact solution. In this paper, we propose a new method to perform the augmentations. It is based on estimated values of the augmented Lagrangians. At each augmentation the converged state is used to extract some data. Such information updates a database used for the Lagrangian estimation. The prediction is primarily based on the evolution of the constraint violation with respect to the evolution of the contact forces. The proposed method is characterised by a noticeable efficiency in detecting nearly exact contact forces, and by superlinear convergence for the subsequent minimisation of the residual of constraints. Remarkably, the method is relatively insensitive to the penalty parameter. This allows a solution which fulfils the constraints very rapidly, even when using penalty values close to zero.