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Now, if you’re from the southern Appalachians, as I am, the word rhymes with “latch” no matter what you’re talking about. People from north of the Mason-Dixon–including the northern Appalachians–and elsewhere rhyme it with “nation,” no matter what they’re talking about, which I reckon is their business. In other words, it’s not a contextual difference, it’s a regional one. I’ve never heard anyone try to split the difference. Perhaps Yankees can say “Appa-latch-an State” out of courtesy and continue saying “Appa-lay-chan” or “Appa-lay-shun” out of tradition, but that’s just confusing for everyone.
To clarify, I e-mailed Jane Nicholson, who wrote:
Via “On the Naming of Appalachia,” a paper by David Walls published by Appalachian State University Press: Be that as it may, Wells’ research along with the Oxford English Dictionary’s entry suggests — to me at least — that early explorers said the word with a “latch.”