I sometimes hear from people who have noticed that, in addition to their hair shedding very aggressively, their scalp and hair have become quite oily. Once they do a little research on this, they sometimes find that an oily scalp can be a sign of another hair loss condition called androgenetic alopecia. But, that amateur diagnosis does not completely fit either because androgen related hair loss does not always present itself with dramatic shedding.
So I might hear a comment like: "I have always thought that I had telogen effluvium because my hair loss started after I stopped a medication that I had been taking for a while. However, now I'm noticing anything about my hair Just the shedding. My scalp and hair are oily and greasy and I've never had this problem before. When I tried to research this, I read in several places that excess oil can be a sign of androgenetic alopecia. Telogen effluvium with a healthy scalp when this is something that is new? "
I will tell you my opinion on this, but I'm not a doctor or hair loss specialist. Still, because of my own experience and the research I've spoken, I do have an opinion, but I'd suggest that you ask a doctor if you still have concerns.
I believe that it is sometimes possible to notice a scalp that is more oily when you have telogen effluvium. And here is why. When many more hair follicles switch into the resting or shedding phase, it sometimes creates inflammation when all of that hair is falling out at once. (This is why your scalp sometimes hurts or turns red or pink.) Now, if your scalp gets inflamed, your body can see this as damage. So, it will try to protect itself and lessen that same damage. But what mechanism does your scalp have to prevent itself? Oil. That is why sometimes people who use very harsh shampoos for oily hair will find that this actually makes the problem worse. The shampoo was harsh to the scalp and the scalp reacted by becoming more greasy because the oil is being produced to act as a protective barrier. It is possible that this was the case here. One thing that you can try is a soothing topical like emu or tea tree oil.
Another possible reason why you might be seeing an oiler scalp is the hair care regimen that you are using. Sometimes, when we are shedding hair, we naturally just want to wash our hair less. Or, we wash it so gently, that's it's not quite as effective. This may cause a little more oil than we are used to. Also, as described above, harsh products and topicals which we try to stop hair loss can irritate or dry the scalp which can bring on that protective oil. Finally, sometimes the oils that we are using to soothe our scalp can make our hair appear to be oily.
With all of this said, it is possible to have two hair loss conditions and it is possible to think you have one when you really have another. Androgenetic alopecia with aggressive shedding certainly was not out of the question. That is why a doctor can be helpful. But since this shedding started when stopping medication, it was reasonable to absorb telogen effluvium unless you started seeing more patterned loss or thinning.