Are there other divisions of Earth following beyond hemispheres?
Not, IMO an ES question, but depending on what level of division, and if you are talking about the surface only or a partition of the sphere extending to the core, terms such as spherical sector, spherical cap, and quadrant would be fairly common descriptive terms for parts of a sphere.

Cap would be the portion of the sphere cut by a plain passing though the sphere.  The Hemispheres are caps cut by the plane passing through either the equator or the poles.  Normally a cap would describe an area formed by cutting the plane parallel to the equator, as in the lines of latitude.

A sector would be a cone centered and the center of the sphere and extended to the surface.  Think of a cone capped by a cap as described above.

A quadrant would typically be a wedge, think of an orange wedge, or a part of one.

I do not think any of there really are especially common to use in relationship to Earth though they are valid descriptors.  Lines of longitude and latitude are the common differentiation that describe these divisions, and phrases such as above/below the 45th parallel to describe the halves of the Northern Hemisphere.