Possible allergies? New rash on snout

Hi vets! We moved a month ago and all of a sudden Zeus developed this rash. It has been present for a few days and only developed about 3 weeks after the move. I was thinking possible allergy because it does seem to get slightly redder after playing or walks outside (first pic is day one and second is after playing outside). It does not seem to bother him at all. We did Benadryl after it was pretty red a few times thinking it was an allergy, but it never fully went away. No other changes besides the move. Any thoughts on how to proceed? Thank you!

Hello - I too wonder about an allergy, or folliculitis caused by demodex mites (which often don't itch), or self-trauma caused by rubbing or scraping to releive another discomfort of some kind.  Sometimes brachycephalic dogs can inadvertently cause damage and dermatitis by licking their faces. Because all of these differentials ('possibities' - and there will be more) require wildly different treatments, your vet may decide to start by conducting tests if it isn't obvious.

.... if the cause isn't obvious.  It's well worth going to see them.