D.B.
I would definitely switch her to whole grains (whole wheat pasta, oatmeal, etc) instead of white flour. Sweet potatoes are better than white potatoes - or at least mix them half and half, either mashed or oven-fried in a healthy oil - and be sure they are really sweet potatoes and not yams, which have a much different glycemic index. Try some more protein too - cheese, yogurt, peanut butter - these are all things that will keep her satisfied longer and prevent the spikes in blood sugar that trigger the hunger feeling. Fruit is good but it's also high in sugar. How about avocados? They are high in healthy fats. Fat, protein and complex carbs take longer to digest so you feel more full longer. I agree also that hydration is really important - hunger can also be a mistaken sign for thirst, as another post says.
That said, it's also very reasonable for a young child to eat more than 3 meals a day. It's supposed to be better for adults too! I think it takes about 20 minutes in adults for the "full" feeling to hit - so the fact that she is asking to eat right after a meal is worth looking at. Give her plenty to drink so she's full, and make sure she waits 20-30 minutes. She may forget after that.
I think it's good that she eats slowly and that you are brushing her teeth, although "looking healthy" may not be enough. It may be early to take her to a dentist. Be sure she has water after her snacks/meals to wash away the sugars that will stick to her teeth.
Finally, I would keep a log of what she eats and when - the item, the approximate quantity, and the time frame. Do this for a week or so. Then you will have data for your pediatrician. You should have a pedi you feel comfortable talking to - most have hours when they call people back to discuss things that aren't an emergency or which don't warrant an appointment. You could even send a copy of your food log with a cover letter stating your questions and concerns, and ask that the doctor call you back when he has a good 10 minutes to talk. Ask the office staff when those times are and, if necessary, request one that meshes with your work schedule. If you do something in the next few weeks, you'll have some feedback from your doctor, and some guidelines/goals for the next few months before your appointment, at which time he can evaluate your daughter.
Good luck!