Enviro allergies are WAY more common than food allergies. Common things occur commonly, so start with what's common before rushing to food changes.
Another clue that's it's environmental is that it's his paws that are the worst itch area -- the paws are his contact points with plants and the ground. Grass seems like its becoming an allergen for ever more dogs, and Spring seems to be the worst season for those dogs.
I would start with chlorhex 4% shampoo once or even twice a week following bottle directions carefully. Like this: https://www.amazon.com/Dechra-TrizCh...dp/B00K4X8TF6/
(note the reviews!!!). Vets commonly sell this, or something like it, but it's cheaper online. (Only apply up to the shoulders... never put it on the head, as it cannot come in contact with eyes).
You can also get chlorhex wipes to wipe down his paws each time he comes in, in between his baths.
See if the medicated antimicrobial baths get him some relief. This will remove the contact allergens and knock down opportunistic bacteria that are attracted to inflamed skin, making the itching worse. You'll know quickly if it's going to help. If it doesn't help, I'd talk with the vet about Cytopoint or Apoquel to get some short-term relief.
The vet I've worked with on food allergies told me that the #1 classic sign of a food allergy is a red, inflamed anus. Lift up the tail and check for a "stop sign butt." If it's pink and normal looking, the likelihood of a food allergy is low. I have a severe food allergy dog -- it's a rough, rough road.