One story you could consider is 'Ligeia'. The narrator here is an opium addict, but this is not mentioned until several pages into the story. In pieces such as 'The Black Cat', the narrator’s derangement becomes pretty clear from early on in the intense, exclamatory way that he speaks and the things he says about himself and other people. In 'Ligeia', although the narrator does come across as quite passionate from the beginning, it seems to have a more conventional cause – his adoring passion for a woman. Rather than talking about himself, without really dropping any hints about his own condition, he spends the first few pages in an extravagant praise of the woman he so loves. He raves about her beauty, and so on, but at first there is no real hint that he is unhinged in any way. But then after Ligeia’s death he states quite bluntly, out of the blue, that he is an opium addict.
I had become a bounden slave in the trammels of opium, and my labours and my orders had taken a colouring from my dreams.
True, he mentions before this that Ligeia's beauty has 'the radiance of an opium dream', but at that stage, we have no real reason to suspect that he himself is an addict, but rather that he is using an exaggerated Romantic idiom. However, once he admits, quite clearly, that he is in fact an addict, his narrative immediately becomes suspect. He himself appears to dismiss his addiction almost out of hand: ‘But these absurdities I must not pause to detail’. We wonder if he is hiding something important from us at this point.
As to the supposedly supernatural encounter that the narrator goes on to relate to us, when he apparently sees his dead wife Rowena transform into his beloved, long-dead Ligeia, this might well be an opium-induced hallucination. In other words, he takes on all the hallmarks of the classic unreliable narrator so often found in Poe's work. We can no longer be sure of what he tells us. Indeed, we may suspect the nature of his relations with these women, Rowena and Ligeia, and we might even consider whether or not he may have played some part in their deaths - certainly in Rowena’s case, as he confesses to hating her, and he also seems to develop an unholy fascination with hanging around burial chambers. This makes us peruse more closely the opening pages, where at first reading, he just appeared to be an infatuated romantic type; but now his outpourings of passion take on a more sinister aspect from the beginning.
You might also consider 'The Fall of the House of Usher'. The narrator here certainly appears much more sober and rational than most of Poe’s first-person narrators; it is his friend Roderick who is inflicted with gloom and incipient madness. However, even this seemingly more reliable narrator confesses in the middle of the story that he too is being affected by the melancholy, indeed macabre atmosphere of the house of Usher, and we start to wonder how much his narrative is affected by this.