Can you add a second door to the right?

It's a door and framing issue. You need a contractor, and some money : )

Is this closet for clothes (on hangars), or a mix of stuff? Switching to a double (upper and lower) rod system is pretty easy. It's also easy for an unskilled person (such as me!) to add battery operated lights to the dark side of a closet -- doesn't help access but improves visibility.

I know you said no major renovations, but adding a door isn't a big or complicated job for a someone with basic carpentry skills. So unless there's some other factor that prevents it, you might talk to a local handyman.

Have you looked at the ELFA System from The Container Store? It is modular so you can use different components to maximize the space. Maybe you have one of their stores you can visit or they will work with you online to create/quote a plan.

This is my guest closet with double hanging, drawers and narrower shelves on the sides.

It does sound like you need to expand the opening. However, then you have a potential issue with the flooring. I would save up to do it, though. If you don't have good storage you need to maximize every inch you do have. There are any number of closet systems you can put inside. Do you wish you had more hanging space or do you need shelves? You can always put freestanding shelving unit in under the hanging rod for an inexpensive fix.

What if you run a closet rod or rods front to back on the blind side, perpendicular to the current rod?

IMO too deep so makes organizing the interior awkward. Post some pics here too. In a comment do not start another post. This type of hanging organizers are awesome in deep closets Amazon has quite a few

I'm having trouble visualizing where the door is--do you have a picture? Is it on the short or long side?

We have two skinny closets, one that is now 48" x 82" and used to be 35" x 82" before a renovation, and another that is 30" x 82". One has the door on the short side and one on the long side. Elfa units have been helpful for both, though the one with the door on the long side isn't super accessible no matter what you do (and that door can't be widened without taking needed wall space from a small bedroom). I have contemplated flipping some of that space into the room on the other side, too.