Thank you so much, Sue!
I'll work on learning about the aphid life-cycle again and check what species they might be, to see which ones may be associated with daylilies. The research you shared mentioned that the daylily they tested was infested with "...potato aphids (Macrosiphum euphorbiae..." so that's a great start.
It sure seems like I get one group of aphids under control and, never fails, up pops another! My initial reading is that soybean oils are not only useful for aphid control, but can be found in organic fertilizers as well, if soil tests show it needs amending. One thing leads to another and, whadda ya know, there I am finding out how many safe, accessible, and affordable ways there are to work smarter in the garden.