You were right. That byte is somewhat delayed. The way I see it, you have two options...
CopyByte 0x020370D4 0x02036E50
Which contains the "Now facing" byte, but it's stored as: 0x11 = down; 0x22 = up; 0x33 = left; 0x44 = right;
or...
CopyByte 0x020370D4 0x02036E54
Which contains the current movement of the sprite.
The first option is better.