I certainly think it's a good idea to check for intolerances if you have frequent problems and don't get dramatically better on the EFI diet. According to IBS experts in Sweden the most common triggers for IBS are wheat, eggs and milk. Egg yolks and dairy are excluded on the EFI so trying to decrease or remove the wheat isn't that much more work. Some nutritionists in Sweden recommend trying to exclude gluten even if you aren't gluten intolerant since many people experience a relief on a GF diet and as they put it, the important thing is if you feel better, not if you have a confirmed intolerance as defined by medicine.
Then of course you have to find the balance so that you don't eliminate too much food and so that you don't go crazy looking for just one more thing to try. If you get fed up experimenting I recommend taking a break for a while and then start experimenting again when you feel like it.
As for symptoms of gluten intolerance the ones I have seen listed are (you may have one or several):
*digestive problems, most commonly D but C, A and bloating can also be symptoms *weight loss *fatigue *depression *anemia *vitamin or mineral deficiencies *sore mouth or blisters in the mouth *certain type of blisters *bone/joint ache *difficulty getting pregnant