Food allergies are actually less common than environmental allergies, though ruling them out is less expensive than a trip to a dermatologist, so it's a good idea to do that first. The hydrolyzed protein diets are actually made to have no recognizable protein, because proteins are the compounds most likely to cause allergic reactions. They break down recognizable proteins into forms essentially unrecognizable as proteins by the body, which theoretically shouldn't cause allergic reactions if the animal has a food allergy. You can have a similar effect by using a diet with completely novel ingredients, BUT many OTC foods, even those with limited ingredients, don't meet those criteria, as they are often processed on the same equipment used for other non-allergy foods, causing cross contamination (the same reason why your box of Nerds candy has a warning about being processed in a factory which also processes peanuts!). If you are serious about ruling out/in a food allergy, it may be best to either stick with one of the HP diets, OR look into making your own food with only a few ingredients, though the Tufts University article I linked below talks about the issues with that briefly. It can take an average of 8 weeks to completely rule in/out a food allergy or see improvements after diet change, so you have to pick one and stick with it.
It is far more likely that your dog has environmental allergies, and if so, the options will be to avoid those allergens completely (often hard, if not impossible), do immunotherapy (which can help greatly, but may not completely eliminate all symptoms), or try to limit exposure and manage symptoms as he is exposed and reacts to allergens. In most cases, people end up doing the latter, and it usually involves a combination of shampoos, supplements, antihistamines and steroids/immune suppressants. I've had personal experience with managing several itchy dogs, one of whom wasn't a candidate for either pred or atopica (diabetic- no steroids, had a previous mast cell tumor- no atopica), and the following is just my experience- run anything you want to try by your vet, as they've actually had eyes on your dog and are experts
In cases where relief isn't achievable with your regular vet, or you want to pursue diagnostics they don't offer, a dermatologist may be helpful.
I've had mixed results with shampoos- the idea is that ones with benzoyl peroxide (also found in some acne washes/creams) help to cleanse allergens from the skin/pores; ones with antibacterial/fungal ingredients do the above, plus help to control secondary infections; and ones with aloe, oatmeal, or cortisone help to soothe the skin and reduce inflammation. Many shampoos have more than one of the above types of ingredients in them, so may have multiple potential benefits if used consistently (I also work for a groomer, and people expect to drop off their dog with horrible skin, have it get one medicated bath, and have it be cured- lol). With my previous allergy dog, I think I tried most types of rx shampoo available at the time, and I noticed that she would have brief (a couple days, usually) improvement after shampooing with the BP shampoo, but her skin/coat tended to be dry after, and she would be itchy again rather quickly. Other shampoos didn't do much for her either way, and some seemed to intensify her itching due to the drying effect. I've had decent luck with my current allergy dog using a shampoo that has benzoyl peroxide, a couple other things, AND (I feel like this makes the difference) an additional ingredient which helps to restore the skin barrier- it doesn't dry him out, which is a major problem for him even when his skin is good. When he's itchy, I bathe him in it at least weekly, and it does seem to help. If the shampooing helps, I would keep it up- if not, ask your vet if they feel something else would be a better alternative.
Supplement wise, I've used fish oil/omega 3 fatty acid supplements with all my dogs, and higher doses with the itchy ones. There is a lot of evidence that it helps, but I've never used it alone to know- it's good for so much stuff that I just give it consistently. Dogs can be allergic to fish/fish oil, but I don't think it's super common. It can take some time to see improvement after starting it, so give it time to help before dismissing it even though he's still itchy. My boy has a glossy, beautiful coat, even when he's itching, and I credit the fish oil with that
My vet has told me that antihistamines tend not to be as effective in dogs as they are in humans. That is consistent with my experience
My diabetic dog was on numerous antihistamines over her life, and some helped more than others, or one would help for a while, then become less effective. My current itchy dog has been on a couple different ones, and we've found one that seems to help him pretty consistently. It can be a process of trial and error. I have found that they work best if started at or before the onset of symptoms (at my current dog's first scratch or even if I notice his skin looking drier, I'll start his antihistamine)- if I wait until he's "on fire", it's much harder to get them under control. They can be used in conjunction with steroids if need (not sure about Atopica or Apoquel, as I really have limited experience with them) to help keep the steroid dose minimal.
I try to avoid steroids when possible for my dogs (and in one, they weren't even an option due to her diabetes), but if other things aren't working, they can help a dog not to be miserable, which severely itchy dogs can be. There are risks associated with them, but they are more likely when a dog has been on them for a long time, or on higher doses. My current itchy dog has gotten pred or short acting steroid injections a few times when his itching has been severe, but I try to manage without whenever possible. That said, I know lots of dogs who are on them regularly (and I have a cat who's on pred pretty consistently unfortunately due to alergies/granulomas), or have been for a long time, and have been fine- though I wouldn't take the chance with my dogs unless it was the only option (and in some of those cases, it is). The goal is usually to give the lowest effective dose as infrequently as possible to manage the symptoms if they can't be off it. I really don't know much about Atopica and Apoquel, but I think particularly the latter is considered to generally have less side effects than steroids, though both are still immune suppressants, and have the risks of that type of drug. I do have some experience with topical steroids (for ears and skin), and use those more readily, though even they can cause problems for some dogs which are sensitive to steroids (ironically, my diabetic dog was fine with them, but my parents' dog, who has some decreased liver function, doesn't tolerate even topical steroids well) most dogs do fine, and they can be helpful if just one area has a problem.
Aside from medication, I've had decent success with clothes for the dog (my itchy diabetic dog had set of cotton "pajamas" that even had legs, and she wore that when she was particularly itchy, both to minimize her contact with her skin chewing/scratching, as well as to minimize contact of her skin with the environment), as well as baby socks on the feet so the nails cant traumatize the skin so much. I keep the nails dremeled so they aren't as sharp, then put socks if they itch badly, though to be fair, you also need to treat the itching medically, as no one wants to have a horrible itch they can't scratch! E-collar (cone) if needed if they obsess over one spot, or are chewing/licking to the point of broken skin (thankfully rare for my guys). If they have raw/oozing areas or scabs, check with the vet, as secondary infections are a possibility with any skin issue. Hopefully you see some improvement with him soon, if not, you may want to consider a derm consult, just to see if they have any other ideas.
These are articles I found that explain the food vs. environmental allergy thing well, and some of the potential treatment options: Tufts article on food vs environmental allergies Merck article on food allergies Merck article on environmental allergies