You give the answer yourself : special relativity forbids any perfectly rigid solid, or more quantitatively, give a bound on the elasticity a solid can have ($Y<\rho c^2$). If you have a real solid, with nonzero elasticity, you can compute the speed of sound within this solid as a function of the elasticity/stiffness (see e.g. on wikipedia for the formula). If you move an end of your big stick faster thant this speed of sound, it will compress the stick, and this deformation will take time to propagate to the other end. This question is one of the many non-working way of making faster than light communications. You have many of them debunked here.
To go into "technical reasons", if your stick is made of atoms, since the atoms see each other through electromagnetic interaction, there is no way the move of a bunch of atoms of your stick propagates to other of atoms faster than the speed of electromagnetic force. Of course, this is a technical reason, which is not valid if your "stick" is made by an exotic material where other forces play a key role (for example out of neutron-star mater, where nuclear force are important), but in this case the violation of relativity would come from the force themselves, which then would allow you to build a (too) stiff material.