Can ice freeze? We know that ice is already the frozen (solid) form of water. The question is more like: Can this frozen form freeze further? Or can it become more solid? (for example, by exposing to colder temperatures and/or a higher pressure).
Water can partially freeze but we can think of a solid frozen state. It is said that virtually all ice in the biosphere is ice Ih (hexagonal crystal form of ordinary ice), but there is also Ice Ic (metastable cubic crystalline) that is present in upper atmosphere in very low temperatures (between −140 and −50 °C). [There are also different forms of ice created in labs.]1
Furthermore, scientists predict that there is an out-of-this-world kind of ice that is present at extremely high pressures nonexistent on Earth but in the range of the pressure regimes on Uranus and Neptune (planets whose major components include ice). At this pressure, oxygen-hydrogen-oxygen bonds get squeezed together, forming new shapes. The new ices eventually become metals, but not as quickly as others had previously thought.2

Static crystal structure of a phase of ice at a pressure of 2 terapascals.2
In the end, can we say that different crystalline forms have different solid states? Can ice freeze further by transforming into a different crystalline form?
Further clarification: "More solid" and "freeze further" might not sound scientific but I wanted to ask in that way for simplification. I meant changes in material properties by saying "more solid". For example,change in hardness, molecules getting closer, stronger bonds etc. [It is even said that ice can become metal at very high pressures.]
Also, when ice is changing solid phases, what is happening there? I thought it is transforming into a harder structure in some phases almost like "freezing further"? I thought this might happen when ice changes into another solid phase. That's why I ask if ice can freeze or not.
Note: There is also fractional freezing but it is related to solutions mainly, for example that contains water and alcohol. It is used in freeze distillation processes.

1 Wikipedia - Ice, Ice Ih, Ice Ic
2 http://news.cornell.edu/stories/2012/01/scientists-predict-out-world-kind-ice
 A: 
Can this frozen form freeze further? Or can it become more solid? (for example, by exposing to colder temperatures and/or a higher pressure).
Can ice freeze further by transforming into a different crystalline form?

The ice will remain solid while lowering temperature or pressure but might change in state, or phase, as you mention. But you should use other words than "freeze further" or "become more solid", since that has no meaning.
Freezing means solidifying. That is a word for the transition from liquid$\to$solid. Since that doesn't occour again in between solid phases, "freeze further" has no meaning.
Ice is usually structured in a lattice. Of course if ice would at some point at extreme conditions make transitions to and from an amouphous state, then yes it would be like "liquifying" the ice again (an amouphous state has the same microstructure as a liquid). This would give ice the same type of structure as e.g. glass. If this is the case maybe the exact definition of the word "freezing" could be discussed, as the ice might be considere to "melt" and "freeze" within the solid state. For me this is mostly a discussion of the definition of the word.
As far as I am aware, ice does not take an amourhpous form, but I might be wrong in extreme conditions.

can we say that different crystalline forms have different solid states?

Yes, and those "states" are called phases (from Wikipedia article):

As you mention, different phases are different molecular structures or lattices. Both temperature and pressure have influence on the points of phase change, as shown in the figure.
The figure could more fittingly be extended to a 3D-coordinate system with a third axis of specific volume. Volume, pressure and temperature are three key proporties that define the points of phase changes, both between gas-liguid, solid-liquid and solid-gas and between different solid states.
