I would open up the entire space and then install some Ikea Pax wardrobe stuff to make this functional. These pics will give you some ideas.

In the recesses, place hanging rods from front wall to back wall approx 16" from each side wall. Do double hang on one side long hang on other side. This will put the hanger ends near the door opening. Do 12" deep shelf stacks against back wall starting approximately 30" from each side wall. Put double hanging rods between shelf stacks. If more hanging needed, substitute hanging rod for lower shelves in either or both shelf stacks. This will maximize usability.

my daughter's closet has the same awkwardness but with doors the open out. This does help with back of the door storage in shoe pockets. could you widen the opening and add wider doors that open out? Could the tower of hats be put on a extra shelf added above the current shelf?

Opening the wall would require a header (and a permit here). If the closet recesseseseses to the right (too many es's) there is not much to do except as directed by the Closet Experts, To the left you can open the wall and add large doors, more doors or lots of doors. 142" of closet is a big closet, same amount of storage as a walk in closet. Having storage space that you can actually use does make life a little easier. This is a task that requires working out a solution to the problem.
A second solution may be to add the closet to the adjoining room. Many times there is a room adjacent to this room that the closet could be opened into the other room. This is a long shot but worth considering.

If this is not a supporting wall then it can be removed and if you go with Pax no header would be required because it would just be a flat wall. that the cabinets go in front of.

i would do what Closet Experts is suggesting, or i would install pull out drawers or shoe racks there and shorten the hang space accordingly so that i could access everything.

i like CE's suggestion of the hanging rods best, though.

Another option would be to build shelves in the area behind the walls that run front to back (similar idea to Closet Experts, but instead of rods - do shelves. You wouldn't have to remove the present rod, just build around it.

That storage unit you currently have with hats in it, could you place it on the current shelf in the closet, but on its side. That way you could put all the hats in the cubbies, then all the stuff that is presently on the shelf could be put on top of the storage unit - you have doubled the space on the shelf and removed the unit from your bedroom.

This is why I LOVE Houzz!! All these wonderfully helpful suggestions and advice and I thank you all.
My "enginerd" husband (minus the pocket protector) not only loved CE's advice but was able to readily visualize the concept and explain it to me over breakfast this morning. Now we just need to get out of the yard long enough to start this project.
Thanks again to all of you!

You have to take advantage of the weather when you can!