No.
Firstly, from a practical standpoint, evanescent waves can only penetrate a few wavelengths. Your proposed cascade of slits would impart attenuations of hundreds of decibels per millimeter. If if you could launch all the photons in the observable universe at such a cable, the likelihood of any getting through would be "macroscopic-thermodynamically" small. Which means simply none would get through.
Secondly, if you mean by:
ome large number of photons could be shot into the cable in the hope that some photons by chance would tunnel through each slit, each time the signal gaining a slight edge on causality?
that one might achieve faster than light signalling through frustrated evansescent waves, then this is impossible.
Moreover, this is not only a case of saying that "special relativity forbids faster than light signalling if we want causality". Maxwell's equations lead to hyperbolic wave equations with a delay conforming with this signal speed limit if we remove advanced wave solutions and this assertion applies whatever field we may be dealing with, whether evanescent or propagating. There's nothing special about "evanescent", even though the group velocity may exceed $c$: the overall delay still respects $c$ and this is one of the few cases where group velocity is not an accurate measure of pulse speed.
Even if we are talking about theoretical tachyons rather than light, we're still talking about hyperbolic propagation equations that respect $c$. For tachyons, this fact plays out in a surprising way. Either the tachyonic wave is (1) not localized, has infinite spatial extent and can travel faster than $c$, or (2) it is a truly localized pulse and the properties of the Fourier transform mean that the pulse's edges are *constrained to travelling at $c$ or slower, even notwithstanding the tachyonic nature. In the truly supraluminal case (1), we can't use such a wave to transmit information as the pulse has no beginning and no end: we can't produce such a pulse! See my answer here as well as QMechanic's here and Jon Baez's exposition here for further details. So even in this case, the structure of the underlying wave equations forces us to respect $c$!
