Replacing eyeball recessed light with "regular" recessed light for shower

Hi folks, we bought a house with a rusty eyeball recessed light in our stand alone shower. I'd like to replace it with a "regular" recessed light with a glass trim rated for wet locations. When I took out the eyeball light fixture, I was left with a can in the ceiling with a female "Edison" screw-in receptacle connected to a black and white wire (presumably coming from a junction box).

I'm not sure what my options are at this point as most recessed lighting ("baffles"?) I've seen online are fairly large and have a connected junction box. I would ideally like to replace the eyeball fixture (removed already) with a light fixture that will fit into the existing can and that can connect to the Edison connector or to the wires (using wire nuts or other connection). But I'm not sure if that's possible or if the whole can needs to be replaced.

I know a picture would help and I will post a pic or two later tonight but any input based on the info i provided would be appreciated.

Thanks!

 

What you removed is called the trim. The part still in the ceiling is the housing.

Look inside the housing and all over the trim for "make-and-model" information. A different trim to do what you want may be available for that housing. If so, that would simplify making the change.

Unless you take pictures that show the ID of the fixture, they may not help us much.

One question: When you look inside the housing, does it appear to be a smooth cylinder or cone, or so you see some slots - usually three - equally spaced around the housing just above the ceiling?