The main thing to visualize is the Hopf fibration of $S^2$, its suspensions, and their various compositions.
Let $f \colon S^3 \to S^2$ be the Hopf fibration.
When you suspend $f$ to get $g \colon S^4 \to S^3$, you effectively embed a 2-sphere as the equator of a 3-sphere and extend the mapping in parallel to 2-spheres of longitudelatitude. Thus away from the poles you still have circles as preimages.
You can see that $f$ and $g$ compose to give a map $h \colon S^4 \to S^2$. To get a sense of how this looks as a fibration, you can work backwards. First, the preimage of a point in $S^2$ under $f$ is a circle in $S^3$. As noted above, each pointwise preimage of this circle under the suspension $g$ is again generically a circle. When the different circles fit together cleanly, it looks like you get a torus fibration, where the tori twist and interlink within each longitudinal latitudinal 3-sphere of $S^4$ analogously to the meshing of circles in $S^3$ for the Hopf fibration. If you now suspend this situation, you get a torus fibration over $S^3$ that looks like $h$ within each 2-sphere of longitudelatitude.
(I'm still not happy with this description but decided to post it in the hope it might spark some ideas.)