Sphere mapping may be used to approximate effects other the specular reflection. Any effect which is dependent only on the surface normal or other single vector can be approximated, including Phong shading and refractive effects. You can use your sphere map to store the outgoing color and intensity as a function of the normal. If you use a sphere map indexed by something other than the reflection vector, you will need to perform your own texture coordinate computations.
When computing your specular sphere map, this color was determined by firing a ray which had been reflected about the normal. To compute a different type of sphere map, determine the color using a different method. For example, to create a Phong lighting map, you can take the dot product of the normal direction and the direction to the light source.