Tentative suggestion: put up could be sky (as in cricket or tennis - a bit weak, I know) and foot of slime in a down clue would be E (although "slime" isn't being written downwards; SKYE is).
I'm happier about the definition. An "inch" is also a small Scottish island, and Skye certainly isn't small. I've only passed through on the way to Harris, but from crossing the bridge to the ferry terminal at Uig was 50 miles!