Sending flyback current to another inductor using crossed diodes
An idea...

If the diode anodes weren’t tied to GND, what happens? You get a large, negative voltage through each diode when each transistor turns off and the coil flux collapses. Add a capacitor to capture the charge and you just made an inverting boost converter.

Rearrange things a bit: put the npn’s on the low side and connect the coils to the supply, flip the diodes to make it a positive boost. This could be fed back to the supply through another diode maybe, or used for another purpose.