The problems with blocked pitots often are quite obvious and clear cut. Be very wary about the static source which is the root of many evils which often are confusing and anything but clear cut in troubleshooting from the cockpit.
It is a good idea to include a detailed and definite inspection of the static and its surrounding area in the walkaround.
The most obscure example I can recall was a tale of a US military bigbird which had had a mod done well upstream of the static vents. Potential undesirable effects on the PEC were not considered, or considered and discounted ... and were not discovered until during the first takeoff ....