You didn't mention it, but did your vet do skin scrapings for mites? Also consider secondary bacterial skin infections caused by scratching.
When you change foods it takes a full month.
If you keep changing every couple of weeks, it's two soon.
If it is in fact food allergies, it's trial and error until you finally correct food. Veterinarians laugh at me. As I would much rather have a dog with mites, then food or environmental allergies. Mites are very easily treated, and allergies are lifelong.
I hope it's something simple, and not SA.
Not all those who wander are lost.
Life is just a leap of faith.
Spread your arms and hold your breath and always trust your cape.
Two things define you. Your patience when you have nothing, and your attitude when you have everything.