In this episode, there is eventually an explanation for why a spider ends up in the apartment next door but one to Yas's, but was there one for why webbing and a large shedding was found in Graham's attic? Or is the implication that there were, and remain, lots of these spiders around the city?
Later in the episode, the Doctor explains that "some stayed here, some went out into the city... hunting".
So that's why there were spiders around the city. How they got there is a bit of a mystery, and not explained very well.
When The Doctor discovers the toxic landfill under the hotel she describes it as "a blocked-in site" - well, it clearly isn't blocked in, because they just walked right up to it. Yaz's mum says that "bits of this is leaking out above here. It's in my kitchen". Again, how can solid waste "leak"? That's not how we were told it got into the kitchen; we were told Yaz's dad brought it up there.
It is suggested that the spider in the flat next to Yaz's got in through the rubbish chute. What I think we were supposed to assume from Yaz's dad finding bits of rubbish around and believing it is a conspiracy is that if bits of the waste were getting out of the landfill and into the city, then so could the spiders that were amongst it.
The Scientist talks about how the Spiders all around the city have been behaving strangely and the Doctor talks about how the Spiders are going out hunting so the premise is that the Giant Spiders have spread throughout the city. In addition in the episode the Scientist states that Spiders like dark, unoccupied locations so Grahams house meets this as the suggestion is this is the first time anyone has returned to the house since before the funeral as opposed to a busy house with people coming and going. Finally Graham with his recent experiences knows to go to the Doctor and tell her where as other sightings will have been called in to police, pest controllers etc.