I think it is a good idea that you have eliminated dairy to start with because its digestion is affected by the damage done by gluten in the small intestine.
When people say they are silent celiacs, some of them mean that they do have absolutely no symptoms. Others recognize that they have been having symptoms which they always thought were 'normal' until they no longer have them. Most can identify at least something that improves with the removal of gluten (besides the damage to the small intestine, that is).
It is important while you potentially have what is known as leaky gut that you do not eat the same thing over and over, because leaky guts allow partially digested foods into the blood stream and then your body can start making antibodies to them too. This is why a rotational diet is often recommended. Rather than being obsessive about it to start with (life is hard enough learning the diet) I would strive for as much variety of food as you can. Now that you have given up gluten you can experiment with some of those vegetables from the produce section that you never quite knew what to do with -- a quick google with its name and gluten free recipe will give you ideas on how to prepare it. You will find new favorite foods in the process.
You may or may not develop additional intolerances. This is not the time to go looking for them! It is time to learn the diet and explore new foods. Try not to use too many gluten substitute processed foods which will contain grains you will not be used to digesting and often contain more fats and sugar than their gluten counterparts. Have fun experimenting with your cooking. You will need some gluten free pasta, bread, and baking mix. I usually recommend Tinkyada, Udi's and Pamela's for beginners. And you can expand your repetoire from there.
I would not eliminate corn or soy right off the bat, unless it starts becoming obvious that they are a problem. I would hold off on oats for now, even certified gluten free oats, because some celiacs are known not to tolerate them either. After about three months it should be pretty obvious if you have additional intolerances and that's where the food and symptom diary come in. This is just my suggested timeline.
"Everything that can be counted does not necessarily count; everything that counts cannot necessarily be counted." - Albert Einstein
"Life is not weathering the storm; it is learning to dance in the rain"
"Whatever the question, the answer is always chocolate." Nigella Lawson
Caffeine free 1973
Lactose free 1990
(Mis)diagnosed IBS, fibromyalgia '80's and '90's
Diagnosed psoriatic arthritis 2004
Self-diagnosed gluten intolerant, gluten-free Nov. 2007
Soy free March 2008
Nightshade free Feb 2009
Citric acid free June 2009
Potato starch free July 2009
(Totally) corn free Nov. 2009
Legume free March 2010
Now tolerant of lactose
Celiac.com - Celiac Disease