I had a girl with similar problems. In our case, she also had little red spots that I would notice while drying her after a bath. She scratched and upon closer examination I noticed some hair loss. The vets could never diagnose the problem, but I truly believed that she had some kind of allergies. Her eyes would get red and runny at times. The grease was the most noticeable, though, as like you said, I could bathe her and the next day she was greasy. Our vet did put her on a steroid for a short time, which did clear up the eyes and helped with the itching and then we had a special shampoo that I used. We also gave her a derm cap twice a day in her food. I bathed her once a week. You didn't mention any other symptom besides the greasy skin, so maybe a good shampoo and some derm caps might help.
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