Eel is often (hot-)smoked not with hay, but with straw in Scania, the Southernmost province of Sweden. It's called "Halmad Ål". I haven't made it myself, but I've tasted it, and it's truly a delicacy! The straw gives an extraordinary and particular aroma. I made a google search and came up with these two recipes. Not true recipes, but sufficient info to try it oneself. The below is GoogleTranslated from Swedish. I have only corrected the most obvious errors. I think this method would be worth testing with meat too. Why not pork belly or beef brisket?
1. Skinned fresh eel is baked on damp rye straw. Served with creamed potato. Use green rye straw. You get a special spicy flavor of the eel and the straw absorbs excess oil.
2. A now quite rare cooking method is "strawing" of eel, which is done as the eel, after cleaning, is put on a straw base, and then cooked in the hot oven. The straw provides a very special taste. One can also vary with e.g. juniper, or other wood.
PS Oh, by the way, contrary to the recipes above, the eel should be brined/cured first, but I guess you figured that out.