ADVICE! Oily Westie, Allergy Health

Good Day!

I have an amazing 10yo male Westie with an excessively oily coat. The vet says it's a reaction to the histamine in proteins... They put him on a diet of prescription Hydrolyzed dog food (Purina Pro Plan Veterinary Diets HA Hydrolyzed, chicken). These issues seem to follow a bad belly ache a few months back.

Does anyone have any experience with this? The new diet seems to lessen the oiliness of his coat but its only been a few weeks. Any tips or suggestions or experiences with this issue would be great appreciate!