What should I do with a shallow closet?
I had a closet once that wasn't particularly shallow, but it did have 3 closet bars mounted front-to-back, and I found the multiple bars very convenient for organizing my clothes.

The left and right bar ran between the front and back walls (the closet had a swinging door in the middle, not a full-width sliding door), and the middle bar's front end was suspended from the shelf above by a triangular wooden bracket (perpendicular to the bar). The middle bar was half the depth of the closet, so you could walk in and reach the side bars easily. Note that the shelf was supported by wooden rails fastened to the wall on three sides, not just shelf brackets on the back wall; the whole system felt quite solid.

I think a design like that would work very well for the closet you describe, and all of the parts were wood, so they could easily be made to fit.

If you are interested then I could make a drawing of the design as I remember it, or perhaps even arrange for a photograph.