From what I understand, the driver chip itself will drop the voltage anywhere between 2-5 Volts (depending on current) due to inefficiencies. I think you can find this on the datasheet under “Total Drop” in the Electrical Characteristics section. There are other drivers that have less voltage drop like the TB6612, but I believe all of them will have some drop.
Edit: I guess that didn’t really answer the “how”. Besides the obvious of just using a higher input voltage, or using a more efficient driver chip, you could also look into an H-Bridge relay setup.
Power MOSFET design (similar to the L298 output circuit) would give you silent control but a single DPDT relay and a single MOSFET for power on/off would allow you to control on/off and direction.
How often do you change direction? Relays these days are reliable but a MOSFET design would be a better solution and you could include current limiting if there was a short on the output.
Just curious… How are you going to drive the P-Channel Fet? If via directly through a port pin that will not work (unless your output pin can be configured as Open Drain). If you want to minimize the part count go with a N-Channel, then you can use your port pin.
@ VersaModule - N Channel MOSFETs need a higher voltage than the supply voltage to switch on (charge pumps come in handy here). ST Micro makes one that apparently doesn’t need that.
This is why the L6470 requires those schotty diodes; to be able to tun on the mosfets.