Reduced-order dynamical models play a central role in developing our understanding of predictability of climate irrespective of whether we are dealing with the actual climate system or surrogate climate-models. In this context, the Linear-Inverse-Modeling (LIM) approach, by capturing a few essential interactions between dynamical components of the full system, has proven valuable in providing insights into predictability of the full system. We demonstrate that Reservoir Computing (RC), a form of learning suitable for systems with chaotic dynamics, provides an alternative nonlinear approach that improves on the predictive skill of the LIM approach. We do this in the example setting of predicting sea-surface-temperature in the North Atlantic in the pre-industrial control simulation of a popular earth system model, the Community-Earth-System-Model so that we can compare the performance of the new RC based approach with the traditional LIM approach both when learning data is plentiful and when such data is more limited. The improved predictive skill of the RC approach over a wide range of conditions -- larger number of retained EOF coefficients, extending well into the limited data regime, etc. -- suggests that this machine-learning technique may have a use in climate predictability studies. While the possibility of developing a climate emulator -- the ability to continue the evolution of the system on the attractor long after failing to be able to track the reference trajectory -- is demonstrated in the Lorenz-63 system, it is suggested that further development of the RC approach may permit such uses of the new approach in more realistic predictability studies.