Can a 'hyperspace' dimension
If we take it literally on the word "dimension", it could be your world has an additional semi-conventional space dimension. That is, everybody is embedded in three space dimensions that resemble our universe, but actually there is a fourth spatial dimension that can be acceded by machinery. So, no, you cannot look in that new direction... yet, it is possible to make machines that move in that new direction.
matter passing through from our world can't be present
That looks like a contradiction to me, if matter can't be present, then matter should not be able to enter. Let us tweak it by saying that matter passing through from our world can't stay (I think that is what you mean).
and is very quickly 'dumped' out of that dimension in another point in space.
Alright, I think we can solve this with topology. Let us say that the three main spatial dimensions are flat at boundary conditions (that is, they can still have curvature, but only locally). Yet, that is only when we see these dimensions at the default position on our new fourth spatial dimension. When we consider the 3D slice of conventional space at some other position on that new dimension we find global curvature such that a straight path (think: geodesic) will return objects to the default position.
That also explains why there is a default position for this extra fourth spatial dimension: it is a stable equilibrium, small perturbations bring objects back right away. Note: this implies that there is nothing special about going on one direction or the other in the extra dimension.
Addendum: If it weren't symmetric on the new dimension, this model would suggest that everything is perpetually moving in a certain direction in that additional fourth spatial dimension making it work more like time in that it goes inexorably forward, then we are talking time travel... well, I suppose that FTL travel is time travel anyway... hmm...
This also has the consequence that you need to consume energy to stay out there, otherwise you are thrown back to conventional space. Now, how far it will throw you depends how you move, both on conventional space and in the extra dimension. So you could travel the same distance by going very fast and no use of the extra dimension, going not so fast and doing a shallow leap in the extra dimension, or going slower but diving deep in the extra dimension.
perhaps due to this other dimensions 'different shape' relative to our own
Ah, you well onto it already.
However, I am not really sure on what the shape is. On one hand, the more condensed space is, the more likely things will go there, suggesting that space out there is “wider”, on the other hand if more direction lead to conventional space, that suggest that space out there is “narrower”.