I don't have an outlet in the closet to power a rope type light. I would like to replace the fixture but maintain the current box The light just isn't bright enough. I did see motion type rechargeable lights but not sure if I would like them as most turn off after 20 sec or so.
You may not have an "outlet", but you DO have a fixture of some sort that powers your current bulb.
Easy enough to replace to what's needed.
In the upstairs 'room' over my shop, I have flat square LED lights that stick down about 1/2" from the ceiling. They are stricly surface-mount, so their power cord is visible. The small wall wart that powers them is plugged into a duplex receptacle in the ceiling, which is powered by a switch near the stairs. Might be something that could work for you.
As a finishing touch on our remodeling project, my son installed lights and switches similar to what
@walta suggested. Did one set in the utility closet that has pull-out drawers (used to be the pantry cabinet) and broom storage, and another set in the coat closet. Works very nicely.
Regarding the motion-sensor lights: Yeah, most of them turn off after 20 seconds or so, but the ones that I have turn off about 30 seconds
after last movement was detected. As long as you are still in its range, it will stay on.
.