I doubt the antiboitics caused the Hiatel Hernia, so if this all started after the antibiotics. Your GERD is likely caused primarily by your gut flora, combined with the hital hernia.
Mine begin after I stupidity went on drinking binge one weekend, and through up. I think that disrupted my gut flora big time. I too was diagnosed with a hiatal hernia, and silent reflux.
I have had symptoms for roughly 20 months. I have a much better control on it, and do not require medication.
Since I likely had the hernia prior to drinking, I assumed I needed to work on fixing the bacteria in my guy. I really do not think there is a magic pill to do this, so you should stop looking. You need to make a permanent lifesyle change.
Cut out all processed foods, and sugars. Eat whole foods, lots of veggies, lean meats (salmon, chicken, etc). Limit yourself to 1 piece of fruit per day. Avoid all alcohol early on, especially beer. I even eliminated dairy and gluten (most gluten is in ALL processed foods anyway).
Find a good probiotic, or make 24-hour probiotic yogurt.
Above all, reduce your stress. Practice your faith, read, work-out, walk after eating, yoga, etc. Get at least 7 hours of sleep each night. Trying to fix your gut can be stressfull, so eliminating stress is a lot harder than it sounds. I've found my faith in God has been my greatest ally.
Drink water, especially between meals, which should help minimize your GERD symptoms. Drinking too many fluids with your meal can make GERD symptoms worse.
As for strengthening your LES, there are breathing excersices which claim to help do that. I've heard there is evidence of this, but it is not a quick fix. Most evidence suggests it takes up to 6 months before noticing. You need to search the internet for the technique, but here is what I know. You need to do it for 30 minutes each day. Take a deep breath in, and breath into your belly (belly comes out when breathing inward), and while holding your belly out, breath out. In theory, this forces you to only use your diaphram to breath out, without your belly muscle assisting in the process. I also do push-ups and burpies to help strenghten my core. My brother thinks that doing squats with a good technique is the best way to strenghten the core. I plan to try that as well. Also, make sure to use good posture, and resist the urge to slouch when sitting. Good posture helps strengthen your core. Sitting is the worst thing for your core...
I still have symptoms, but they are nowhere near as bad as they used to be. I can function throughout the day, and I don't begin to show any symptoms until the afternoon. Some days I go almost symptom free, while other days are more noticable.
Measure your sucess by weeks, and not days, and NEVER give up!