The schematic you show has the !HI and LO inputs paired together. This means that you only need one signal to drive each phase, but that either the top or bottom FET in each pahse will always be conducting. This is commmon with a sinusoidal drive, where each phase is fed a continuously varying PWM to approximate a sine voltage at the motor terminals, but as you indicate that you want to have one phase floating, I assume you're aiming at a trapezoidal drive - which you can't achieve with that arrangement. You'll need to control all six inputs independently.
For each possible arrangement of Hall sensors, which can be disposed in various ways around the stator, and winding (phase windings can be connected either way into the wye or delta pattern, reversing the physical direction of rotation) there'll be a table of required switchings - this should be available from the motor manufacturer.
This is taken from the datasheet for the DRV8307 showing a typical table. Since your HI inputs are inverted, you'll need to invert the signals to them.
The note (1) about synchronous rectification is a bit of a misnomer (that's used in switchmode supplies) but operates the same in inverters - it switches on the FET that would otherwise have current circulating through the body diode, which reduces losses and improves efficiency.