4" is not nearly enough clearance so you are on the right path seeking more. I had about 6" and it wasn't enough...now I've got about 8". Much better.
Something to keep in mind. If you are already 13'6" tall, you are at the height of big rigs. Taller and you will possibly have other issues.
I had a subframe constructed under ours which raised it 2" and now we are at 13'3". Before I did that I hit a bridge once and even going slow the damage was done. I DON'T recommend it!
Do you know the actual height of your trailer? Not advertised height, how tall it really is? For off-road and uncertain clearances, I bought some pvc pipe and a couple joints and can assemble a measuring stick for if I'm ever in doubt.
If you haven't already, suggest you get a trucker GPS and trucker atlas and plan your routes with them. I've also got an AllStays app which has clearances in it. I didn't have these when I hit the bridge.
If you never vary from the same route, you probably don't need the above, but if you're going to go to new places, I think you'll appreciate them. Cringing is not good, but I can relate...big time.
2009 Ram 2500 CTD
2012 XLR Thunderbolt 300X10