How is the current flow perpendicular to the wire? This answer gives a great explanation of how surface charge builds up to force the current to move perpendicular to the wire: https://physics.stackexchange.com/a/102936/41086
However, it fails to address the magnetic field. Moving charges generate a magnetic field, so in addition to the electric field creating a force on the charges, so does the magnetic field.
The answer doesn't explain what effect (if any) the magnetic field generated by the moving charges in the transient state has on the charge build up or the effect of the magnetic field in the steady state.
So could someone give me an explanation of how the current flows perpendicular to the wire, by taking the generated magnetic field into account? Essentially, I'm looking for an extension to the answer I linked to above by including information on the steady state and taking the magnetic field into account.
 A: Let's model the real wire as two infinitely thin wires carrying 1 mA of current 1 mm apart.
$$ B = \frac{\mu I}{2 \pi r} $$
$$ F = q v B $$
Plugging this in, and considering the speed of the current being one third of the speed of light (close to the actual value, depending on the wire) we get that the acceleration exerted on each electron is of the order of $10^{12} m/s^2$ Calculation in Wolfram|Alpha. That is a quite high number, so my first model looks quirky.
We need something better. Take the section of the wire, the current flows on the surface. You want to see the magnetic field at the topmost point produced for each point in the surface. The first thing you notice is that, for each point, the one in the oposite side will have a magnetic field that will partially cancel each other. If you look at the wire from the direction where the  current is going to, only the component to the left will remain. NOTE: the closer to the top, the more intense this would be, but also the stronger the cancellation.
Now, where is the magnetic force pointing at? Towards the centre of the wire. So, as electrons flow, they will be pushed towards the middle, but also that would bring them closer, and therefore augmenting the repulsion. 
So, what happens in the end? If the wire is small enough, the magnetic forces are strong enough so they can push the electrons in, and the current flows all across the body of the wire. If you make it thick enough, the stronger components in this force, the ones coming from the opposite side of the wire, will be further away, so the flow will remain in the surface. That is one of the reasons for thick wires to be made out of many small ones bundled together. The others are: it is easier to bend, and if there is one defect (a crack, for example), it will only break that fiber, but not the whole cable.
A: For slowly moving charges, magnetism is just a relativistic correction, so the relative size of its effect is $O(v^2/c^2)$. Since $v$ is very small for charges in a wire (less than 1 cm/s), the effect will be insignificant. 
Since parallel currents attract, the current will be attracted to the center of the wire a tiny, tiny bit.
