Would it be correct to merge with hyphens one-third-contiguously in the following phrase?
No. "One-third-contiguously" would indicate things that are only partially contiguous, which means that they would abut at something less substantial than a mathematical point (the minimum requirement for contiguity), and the whole phrase being used adverbially. Perhaps if you were to put it into context someone could suggest a wording less damaging to Broca's area.
If you want to modify "one-third" with an adverb, put it in front with no hyphen:
That's nearly one third of the total.
I've eaten easily one third of the cake.
We're roughly one third of the way there.
Pay me exactly one third of the total.
However this tends to modify the amount, or how confident one is about the precision; I'm not sure exactly what you're saying but using "contiguously" in this position would sound strange and the meaning would be unclear.
You may better off using an adjective instead:
each contiguous third
but inserting this back into your original sentence makes it clumsy; depending on your precise meaning something like
per contiguous section making up one third of the time zones
may be your best bet.