Vitamin and zinc is a good place to start, but keep in mind that if you could be sensitive to sulfur due to genetics or some kind of deficiency in the body. Vitamin B6 is one supplement that makes my hair loss way worse. Even activated version, p5p does not work for me. I am not sure why, but I've seen many people reporting the same issue. Same goes for L-glutamine.
I think the safest place to start is to kill toxic microbiomes with antibacterials like mastic gum (good for h.pylori, though it's effect is controversial), coconut oil, cat's claw, and manuka honey. Oregano oil is too strong for most people; it can even puncture holes in your stomach. Antifungals are usually strong and most likely to worsen your symptoms in the short-run.
Next, restore good strains of probiotics in the gut, but watch out for histamine producing strains like L. casei. I would suggest to try Bifido strains in general.
Histamine degrading - bifido bacteria, b. infantis, l. rhamnosus, L. Salivarius, L. sporogenes, L. gasser, b. longnum
I would avoid sulfur supplements in general, even though some people have success from this. If you are desperate to try, start from low dosage and work your way up GRADUALLY.
If you have any questions, PM me or leave them in the comments. Good luck.