pressing after sintering may distort the pore size distribution by breaking the skelton connecting the pores.

I figured there might be an effect on the structure.  Thank you very much Ojha!

It depends on how much porosity you are expecting. Probably, you may use some kind of organic binding agent (melassa, for example), then carbon mass can be subjected to compaction, and finally, to sintering. Therefore, you will be able to to achieve different porosity by varying of applied pressure and admixture amount. 

First of all, the serious problem during powder processing is agglomeration. If you have nanopowders or micro-sized powders , the agglomeration is very hard to eliminate, however it  is still possible. It means that before pressing you have additionally to mill ( the best is in attritor mill, wet milling!!) when is possible to decrease the factor of agglomeration ( for 10-40 times depending of powder nature). Other way is to sieve the powders before pressing. During pressing is almost impossible to destroy hard agglomerates and due to that it is needful to have above mentioned operation before pressing. The fact is that if you have agglomerates, you will have it after pressing as well . Probably less because during pressing you will destroy soft agglomerates ,  but not a hard. The result is that you probably  sintered agglomerates, not individual particles and the microstructure will be different.  All other answers are related  to porosity but much more important is particle size, their distribution and factor of agglomeration and their influence on sintering after pressing Naturally, the porosity is connected with above mentioned problems. .

Pressing after sintering will have effect on porosity and porosity will be reduced during pressing. Of course that can be optimised depending on the applied pressing pressure. However optimisation will be dependent of application of filtration. 

I suppose two step sintering will be better. Initially sintering temperature and time may be optimised at lower temperature ( and time also) and after that pressure can be applied for pellet formation. After that final sintering can be considered to have desired porosity.   

In ceramics, the processing method (pressing, sleep casting, etc...) is basically linked to the nature of final product (refractories, porcelain, stoneware, sanitary ware, fired clays, etc...) and the quality of the raw materials. It is not appropriate to give a general answer.

The pressing should then be considered according to the target material

After overcoming the appropriate sintering temperature for the material is inefficient applying pressure; on the traditional processing route compaction in green is used to increase the density and achieve an intimate cohesion of the particles. So I think that the only advantage would apply pressure before or during sintering (as in the SPS), I see no sense in applying press after ..

Thank you everyone, scientific tradition is completely different from manufacturing tradition.  That's a whole other beast, as I am sure most of you understand.  Looks like I have a bit of work ahead of me.  Thank you all for your input.