We seem to have created a problem for ourselves by making Eöl an Avari. He could have tagged along during the long Journey but now we have to invent a reason for him to be in an area where he’s the only Avari among a whole lot of elves. It really seems counterintuitive for him to be in Beleriand, of all places.
Could a solution be in his affinity with magic? Maybe he is always looking for ways to grow more powerful. He’s somehow found the black metal, and wants to create armour and weapons out of it. He senses the power of Melian from afar, and reaches Doriath - maybe he has some kind of dowsing or divining instrument, of just by sniffing his way - and he can’t pass the girdle but finds Nan Elmoth, where he wants to stay and use residual magic to increase knowledge and power.