There are lots of ceiling mount LED lights .

Do you have switches or need a pull string? LED disk lights mount flat against the ceiling and put out tons of light. If you want to go truly easy there is a LED light that just screws into an existing socket. Search on LED easy light or LED spin light

Looks like the photos didn’t load. Lyndee, totally forgot about the pull string, that’s what we have. We will have contractors in for other work, but will have to weigh the cost of wiring for it versus sticking with current pull string. If we do get it wired for a switch, the issue is we have just shy of 5 inches for this light to not hit the wall.

For reach in closets that will be for the occasional guests, BTW.

It looks as though the light is very close to the door, so you could probably replace the pull switch with a push to break switch so that the light comes on when you open the closet. I find them quite convenient, as long as people actually shut the closet doors when they have finished reaching in.

In our closets I eventually replaced the lights with recessed fittings, but until then I used metal cage type lamp shades to protect the bulbs as they were prone to getting knocked as we took things out of the closet.

This Commercial Electric 54606341 "Spin Light" has a motion sensor and screws into the existing ceiling fixture like yours. It should turn on when somebody opens the door or reaches in.

If you want to replace the fixture, Home Depot has a bunch, including this Lithonia FMMCL 840 PIR M4 7 in. White LED Motion Sensor Flush Mount Closet Light. Just check that any light you're considering can be attached to the electrical box you have (i.e., check the required electrical box size).

^^ that motion detector light is very cool and maybe your best option.

for me, I needed my closet to be lit from the front. otherwise the top shelf would just block the light and make it useless. I installed LED strips to the inside of the door frame, pointing forward. the kind of lights you use under cabinets. the switch also got installed inside the frame where I could easily reach it. (although a motion detector would have been so much better!)

to plug it in, you could run the wire up to the top and convert your socket into an outlet with one of these. it's not beautiful, but it would work. or you could get somebody handy to hard wire it and make everything look nice.

Here's another one will a pull chain: Hampton Bay 8 in. 60-Watt Equivalent Brushed Nickel Integrated LED Drum Flush Mount