Half-wave and full-wave rectifiers
You cannot physically connect a half wave rectifier to the output of a full wave rectifier unless you do so via an isolation transformer. A full wave rectifier comprises of two double-diode half wave rectifiers that are "added together". If you try and disrupt this by using a plain half wave rectifying diode circuit you will cause a short circuit.

Using a transformer isolation method would prevent the disruption but also give you no benefit whatsoever.

Feeding the AC through a half wave rectifier then its output through a full bridge rectifier gives you nothing except extra volt drop in the diodes.