grammar - How to modify "one-third" by an adverb?

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.