Old sci-fi book where a man is transported to another world, and meets woman with green skin. Involves black floating transport?

This "green woman" memory might possibly be from Chaining the Lady, the second book of the Cluster series by Piers Anthony.

There was no green woman in the book itself, but there was one on at least one version of the cover.

In this book, Melody of Mintaka is chosen to have her "aura" transferred to another "sphere" to pass along the technology of "Transfer" -- this mind transference itself. She is chosen because her aura is exceptionally strong (as was the case for Flint in the first book); this gives her a better chance to survive the slow fading of her aura long enough to pass along the knowledge and perhaps even be transferred back to her own body.

The first book, however, Cluster, featured Flint, a stone age artisan (named for his working material), who was chosen for the same task, but was initially human. He did encounter a flying machine at one point that was basically a box in which he put his feet.

I suspect you might have read an omnibus edition with all three of the original trilogy in a single volume, and a cover taken from the second book.