Previously, the author described a closed-form analytical approach to obtaining the complete solution set for four-mirror anastigmats in which all the mirrors were spherical. Two novel types of four-spherical-mirror anastigmats with concave primary were reported. This approach has been modified by setting up baseline systems with good first-order characteristics and adding the minimum number of aspheric lenses necessary to provide anastigmatic correction. In this way, searches for useful four-mirror anastigmats with one, two, or three mirrors kept strictly spherical have been carried out. By using this technique, a system with only one conicoid mirror can be found that has very similar first-order properties and equivalent correction to a system with three conicoids.