I don't think 14" is deep enough to hang coats the normal way, but the slide out idea might work.

How wide would the closet be? If it's just the width of the old bathroom door, you won't be able to fit more than 2 slide-out things for coats to hang left to right instead of front to back. My guess is you'll only be able to hang 4 coats on each slider (our winter coats tend to be bulkier than spring jackets) so only 8 coats would fit in the closet. Will you still have room for pantry items?

You could get away with a little less, though, I bet.

Measure a coat hanger; add 2 inches on each side.

There was a wardrobe once in a Martha Stewart book (organizing book) that was converted into an office/desk. But still attached in the back were these hinged hook-thingies.

Here' how it was;

imagine an 8 or 10-iinch-long piece of wood, one end hinged to the back wall of the wardrobe, so they stick out toward the front, but can be swing over (they were hinged to swing toward the outside of the cabinets, so the ones on the left swing toward the left; ones on the right, toward the right). On one side of this piece of wood are wooden pegs, so they point toward the side of the cabinets. That's where you hang the coats.

It looked like a neat way to get a lot of coats in not a lot of space. When you wanted to access a peg, you could swing the board off to one side.