"I was perfectly satisfied to write science fiction knowing that it would pay very little, that it would be seen by only a very few people."
- Isaac Asimov

Hyperspace as usually presented today is as an alternative to "normal space," that being the reality that we humans occupy.

John W. Campbell used it just three years later sans hyphen in Islands of Space:

Well, we would be stuck if we didn’t have this new method, for the acceleration would take too long. However, in the hyperspace we are going into, a new condition exists. In this space, you never saw light standing still, did you? Now, when light is made, it is at rest, and yet it instantly attains a velocity of 186,000 miles a second. How? Light cannot go slower than that speed and exist. However, the presence of certain things in space will alter the minimum velocity. For instance, it can go more slowly through glass, for the presence of the glass alters the condition of space. Well, in this hyperspace we are creating, matter cannot exist at a velocity lower than a certain quantity, and we determine that quantity by using this apparatus. We are subject to an infinite acceleration — and we are subject to that acceleration in every particle. Remember, that we have no weight when we offer no resistance to gravity, but we yield and fall freely. We yield to the infinite acceleration, and are not hurt by it. Result — we move at the velocity we desire,” explained Arcot.

“Well — you go into a hyperspace — and move at any speed you please. I wonder then, how are you going to see where you are going?”

“We won’t. While in that hyperspace I don’t expect to see anything. We will start the ship in the direction we wish to go, then go in that line when we fall into hyperspace, and since we direct our path, we can avoid stars in that way. Meteors, or objects too small to be seen, we need not worry about, for the old law correctly stated, is, two bodies cannot be in the same space at the same time. We won’t be — we will be in a different space. However, the gravitational effect of a star would ruin our apparatus, and drain out the stored energy, and we would materialize perhaps in the center of a white-hot starl The moral of which is, look out for stars!”