When we talk about Occlumency lessons and the memories Snape deposits in the Pensieve, I think most people (myself included) assume that Snape only puts three memories in there. But what if he puts three sets of memories in? As in, each strand of memory he pulls away contains a cluster of memories around a particular subject or train of thought.
Here how it’s described in the book:
Snape pulled out his wand from an inside pocket of his robes and Harry tensed in his chair, but Snape merely raised the wand to his temple and placed its tip into the greasy roots of his hair. When he withdrew it, some silvery substance came away, stretching from temple to wand like a thick gossamer strand, which broke as he pulled the wand away from it and fell gracefully into the Pensieve, where it swirled silvery-white, neither gas nor liquid. Twice more, Snape raised the wand to his temple and deposited the silvery substance into the stone basin, then, without offering any explanation of his behaviour, he picked up the Pensieve carefully, removed it to a shelf out of their way and returned to face Harry with his wand held at the ready.
-Order of the Phoenix, Ch. 24
Snape extracts three strands of memory, but we don’t know what they contain. They’re described only as a “silvery substance… stretching from temple to wand like a thick gossamer strand.” In comparison, here’s how the memories Snape gives to harry at the end of DH are described:
Something more than blood was leaking from Snape. Silvery blue, neither gas nor liquid, it gushed from his mouth and his ears and his eyes, and Harry knew what it was, but did not know what to do - A flask, conjured from thin air, was thrust into his shaking hands by Hermione. Harry lifted the silvery substance into it with his wand.
Again, the memories are described as a silvery substance. There’s no mention of it coming out in bits and pieces, as if separate memories came out individually, but as a single amorphous substance which can be deposited into a flask by lifting it and placing it inside.
The stone Pensieve lay in the cabinet where it had always been: Harry heaved it on to the desk and poured Snape’s memories into the wide basin with its runic markings around the edge.
Here too, where Harry empty those memories into the Pensieve, they seem to be described as a single substance – we know that the flask Harry empties held numerous memories from Snape in it, but they don’t seem to be separated in any way. They’re extracted, bottled, and put into the pensieve in the same way a single strand of memory would be. It seems very possible that Snape could have pulled away a single strand of that non-gas, non-liquid substance at the start of Occlumency lessons, yet it contained numerous memories instead of just one.
The way extracting and storing memories seems to be complex. In Half-Blood Prince, Dumbledore has the memories he’s collected stored and organized in individual crystal bottles. This implies that memories can be extracted both one at a time and also en masse, the way thoughts work - sometimes we think of a single experience, and sometimes we go through multiple memories simultaneously, jumping back and forth between each one. The Pensieve also holds multiple memories simultaneously, as we see in Goblet of Fire when Harry first falls into it and experiences several of Dumbledore’s memories. Those memories all follow the same theme - various Death Eater hearings after Voldemort’s fall.
While it’s implied that the Pensieve shows memories in chronological order, we don’t know if a person viewing them is necessarily seeing the earliest one in a set. It’s possible that a practiced Pensieve user, or one who enters the Pensieve with deliberate intent to view a series of memories, is more likely to view those memories starting with the earliest one. When Harry went into the Pensieve and saw Dumbledore’s memories in Goblet of Fire, he was cautious and curious. We nevertheless have no way of knowing if there were earlier memories he didn’t see, or even if those memories were chronological order, but they seem to be. In contrast, when he goes into the Pensieve in OotP to view Snape’s memories, he’s being reckless and thoughtless. – he’s just had a fight with Cho, he’s been angry for months, and he takes it out on Snape by invading his privacy. It’s very possible that this attitude and approach to someone else’s memories affects how they’re viewed, and that he drops into the middle of a series of thoughts and SWM is just where he happens to land. Maybe it’s exactly because he feels this way that he does, because his own feelings of frustration and rejection connect with those in Snape’s memory.
It’s also possible that the order memories can be viewed in is determined by how they’re extracted and saved – in which case, if Snape was just dumping groups of memories into the Pensieve during Occlumency lessons, their order may not have been chronological, or there may not have been any order to them at all, as opposed to when he gave Harry the memories that we see in the Prince’s Tale, which may have been in chronological order because his life was flashing before his eyes as he was dying. Therefore it’s also possible that Harry dropped into SWM in OotP because Snape extracted a group of memories and the most triggering one was at the forefront. But if Harry hadn’t been interrupted, perhaps the memory would have shifted into other ones of Lily - Snape’s apology to her, their childhood memories, etc.
There seem to be multiple possibilities for how extracting memories and viewing them in a Pensieve works. When Harry sees Snape’s memories in The Prince’s Tale he sees the same one he saw in Snape’s Worst Memory. It’s possible that when he first sees it in OotP he just happens to fall into that part of a longer timeline of memories, or that he falls into a cluster of memories of which that is the most prominent one. All we know is that Snape put at least three memories into the Pensieve, but Harry saw this particular one. It’s possible that it wasn’t just three memories Snape put in there but three sets of memories. If that’s the case, it’s also likely that these sets are organized differently than what we see in The Prince’s Tale. If his motivation in extracting memories was to hide them from Harry in OotP (and by proxy, from Voldemort), and if in that moment he was in a measured and controlled emotional state, it would affect his choice of what memories to share. Perhaps these are a more robust set focused entirely on Lily, and presumably other areas of his life he doesn’t want Harry to have access to, whereas the set of memories he gives Harry in DH are about conveying Dumbledore’s message that Harry has to die and giving it context (to which Snape and Lily’s friendship is integral).
Obviously this is all conjecture and there isn’t much textual evidence to go on. Memories and the Pensieve have been a grey area of canon at best. Nevertheless I find the idea very compelling, that if Snape could hand over a whole series of memories as one unit in DH, it should be possible to extract them in groups as well as individual ones. In which case the question isn’t what specific memories other than SWM did Snape put in the Pensieve during Occlumency lessons, but what sets of memories? SWM may have belonged to a larger set of memories of Lily. Perhaps the other two sets have to do with his history as a Death Eater and might include his work as a spy, and his memories of childhood abuse that are personal and make him feel vulnerable? All Harry sees when he breaks into Snape’s mind is his father shouting at his mother. The extra-textual information given on Pottermore and approved by the author (boo,hiss) says that Snape’s father used to beat him with a belt… if Harry broke into his mind and saw the most vivid memories, would they be of Snape’s father shouting or of him being whipped? Wouldn’t it be the latter? Unless… it wasn’t accessible because it had been put in the Pensieve for safekeeping.