The efficiency of space-mapping optimization depends on the quality of the underlying coarse model, which should be sufficiently close to the fine model and cheap to evaluate. In practice, available coarse models are often cheap, but inaccurate (e.g., a circuit equivalent of the microwave structure) or accurate, but too expensive (e.g., a coarse-mesh model). In either case, the space-mapping optimization process exhibits substantial computational overhead due to the excessive fine model evaluations necessary to find a good solution if the coarse model is inaccurate, or due to the cost of the parameter extraction and surrogate optimization sub-problems if the coarse model is too expensive. In this paper, we use an interpolation technique, which allows us to create coarse models that are both accurate and cheap. This overcomes the accuracy/cost dilemma described above, permitting significant reduction of the space-mapping optimization time. Examples verify the performance of our approach.