Ruth's idea re running tests for tick-borne illness is a great idea. I would definitely do that.
How long has he had this rash? If it's somewhat recent (the last month or three), my next question is -- Where do you and your pup go?
It's spring, which means that everyone -- and I do mean everyone -- is putting funky stuff in their yards and on the lawns. Parks, schools, neighbors. A lot of that stuff is petrochemical herbicides, pesticides, fungicides, etc. And the soft skin on dog tummies and armpits is very sensitive. And that part of the body is RIGHT THERE where all the weird chemicals are.
If this rash just started to appear somewhat recently, I'd rinse my dog well in clear warm water every single time I brought him home from a walk or any trip where there is a lawn or garden. And take your shoes off. You may be tracking that stuff in on your carpet too. Vacuum often. Change his bedding extremely frequently. It might help.
Even if he's not reacting to environmental toxins, it could be an environmental allergy. He could be allergic to a specific kind of grass or weed (or many). Taking the steps I listed above will help.
I have to tell you, I'm not a big fan of shampoos, even veterinary shampoos. Try it for a while, but if it's not helping, you may want to reconsider. (I learned a lesson the hard way on that.) But when you come home from walks and such on a daily
basis, just rinse. You don't want to dry out his skin any more than you need to.
And, as Jean says, seeing a specialist is something to strongly consider if this doesn't clear up soon. Often, we spend a lot of time and money hanging out with our regular vet, trying a number of different things, taking a bunch of different tests, whereas if we had just gone to a specialist from the beginning, it would have likely saved us both agony and financially. So that's something to think about if things don't clear up fast. You'd want a veterinary dermatologist to start out with. They almost always know a lot about allergies and the immune system. And if this is beyond a simple skin issue, they can give you a referral to the right specialist beyond that.