There's no way to modify it on my Nexus 5, but I don't know if the Moto X might have its own special setting. One workaround is to use Power Toggles, which can put the date (but not the month) as a persistent notification at the upper left of the screen.
You can turn on ambient display and not use the moto smart screen. It will show the lock screen with the date under the clock on smaller letters. It will turn the screen on with notification tiles but will use more battery.