The problem with hydrocortisone is that it actually thins the skin. My sister has psoriasis and has been put on hydrocortisone- her skin tears so easy, like a wet paper towel. I really think this drug should only be used as an absolute, absolute last resort.
The skin is one of the detox organs, it is a secondary detox organ. If the body is trying to get rid of a toxin, the urine, breath, sweat, and feces can only do it so fast. So the skin is called upon to excrete the toxin also. Eczema can be caused by many different reasons! internal staph bacterial (or other bacterial) infection , parasitic roundworm infection, celiacs disease (gluten intolerance) - which is common, milk intolerance, nut allergy, a systemic yeast overgrowth, etc.
I would try to change the diet- try something for just one week. Try getting her off both cows milk and wheat gluten. (which i think are the most common allergy). If this is the problem, you should start to see some improvement.
Milk and wheat gluten are fertilizers for fungus. I mean- it thrives on those two things. So there could be that involved also. You can try some antifungals. There is a product called threelac you can try. Or garlic tablets, caprylic acid, olive leaf, and even oregano oil drops under the tounge. Fungus loves the milk sugars the gluten carbs, and all other sugars.