I'm not so sure. Memory isn't genetic, so why would it appear in a clone? Unless we're talking about a molecular clone, a person reproduced exactly, which should retain the structures that encode memories. But in that case you'd retain ALL memories, barring reproduction errors, and the brain spiders wouldn't be needed in the first place.
Brain spiders and soul transfer tech take information recorded from the original brain and plant it in the cloned brain. It's like copying info from a hard drive in one comp box onto an external drive, then constructing an identical box. It'd be impossible for that second computer to have the same information on it even though it's a copy, so you'd take that external drive and upload the info from the first computer into the second one. Lab techs shouldn't worry about memory resurfacing at all.
Note that there's no way to prove memory drift until someone flash-clones themselves. And we also haven't proven that anything beyond simple instincts survive through genetics. C-Man, maybe your character might find he prefers a certain food or favors his left hand over his right, but that's no more than preferences his source was predetermined to have in the first place. He's not going to suddenly remember the origin's first date.