We show how the response of a chaotic model to temporally varying external forcing can be efficiently tuned via parameter estimation using time series data, extending previous work in which an unforced climatologically steady state was used as the tuning target. Although directly fitting a long trajectory of a chaotic deterministic model to a time series of data is generally not possible even in principle, this is not actually necessary for useful prediction on climatological time-scales. If the model and data outputs are averaged over suitable time-scales, the effect of chaotic variability is effectively converted into nothing more troublesome than some statistical noise. We show how tuning of models to unsteady time series data can be efficiently achieved with an augmented ensemble Kalman filter, and we demonstrate the procedure with application to a forced version of the Lorenz model. The computational cost is of the order of 100 model integrations, and so the method should be directly applicable to more sophisticated climate models of at least moderate resolution.