Whether you wake to a dawn-simulating alarm clock or not, continue to expose yourself to light as soon as possible after waking.  This will stop melatonin production and increase your wakefulness.

Throughout the day, get as much light as you can.  Most folks can sneak outside for 5-10 minutes. Run errands at lunch or eat outside. Do as much as you can to get that sunshine.

The more bright natural light you can get during your normal waking time, the more your body will know to gear down at your normal sleeping time.