For the original question, consider the maximum difference between the current state and the desired steady state. When that maximum difference crosses your "close enough" threshold of 1e-3 in a downward direction, stop. Since the maximum difference is less than 1e-3, so are all the others.
For the new question asked by max gilles, the Events function can return a vector. See for example the orbitode example. In that case only one of the events is terminal, the other is simply logged and used to plot maximum distance after the ODE solver has completed its work.
