It's been a while since I read it, but my memory is that within the Deepness in the Sky setting, there are no zones other than what is later referred to as the "Slow Zone" in Fire Upon the Deep. That is:
-There is no mention of the Zones, let alone of the characters being aware of them.
-All of the travel is sublight speed.
-At the end of Deepness in the Sky, Pham Nuwen embarks on a journey to the center of the galaxy. It is later revealed in Fire Upon the Deep that this put him in the "Unthinking Depths". If humanity had been aware of The Beyond and how the Zones work, surely Pham would have wanted to go to The Beyond, not the Unthinking Depths.
We also have the Emergents using people as makeshift AI, and if there were in the Beyond, they would just use actual AI.
wikipedia has this to say:
A Deepness in the Sky takes place in the Slow Zone, though Vinge does not explain the connections, and the characters in the story remain unaware of the zones' existence. The sun's inexplicably strange behavior, the unusual planetary system (with only a solitary planet and several asteroid-sized diamonds), and the discovery of "cavorite" on the planet may indicate the system originated in the Transcend, though it is currently moving outward from the Unthinking Depths. Vinge's characters speculate that the Spiders descend from an ancient star-faring civilization, and that the anti-gravity material and other strange artifacts have connections with that civilization. Unfortunately, they guess the structure of the Zones (though not the actual properties) backwards, coming to the conclusion that the bright center of the galaxy is the most likely location for advanced civilization. This leads Pham on his path inwards to the Unthinking Depths, and his eventual resurrection.
So the implied sequence of events is this: The Spiders' planet was originally in the Beyond, on the other side of the galaxy from where the Qeng Ho and Emergents found it. It then traveled into the Slow Zone, eventually passing through the lower portion of the Slow Zone and possibly the upper edge of the Unthinking Depths, causing a collapse of the civilization. It then came out the other side, continuing on back toward The Beyond. Eventually, its increased distance from the galactic core, along with time for its inhabitants to rebuild, resulted in the Spiders reaching a 20th century Earth level of technology, and the radio emissions of their civilization alerted the Qeng Ho and Emergents to their presence. Pham Nuwen, seeing that the planet had signs of previously advanced technology, and observing it traveling away from the galactic core, incorrectly surmised that the galactic core was a region of advanced technology.