I wonder if you're growing one of the poet's narcissus. Take a look, I think they do well in warmer climates and although there is usually a red rim to the cup there are other small cupped narcissus which don't have that.

It's not poet's narcissus, which I have in the garden, though it looks like it could be a distant relative, with its small flattish cup. It looks more like that than like any other daffodil. The cup is lemon yellow, by the way, and the whole effect though pale is distinctly yellow and white. I looked through the catalog section on poet's narcissus and didn't find a match, nor in small-cupped narcissus.

N. poeticus is native locally, though rare; I knew a woman who had a woodland full of it, but she was the only one. I suspect it's more common up in the mountains than where I live in the hills. In the Alps north of Milan it used to turn the mountains white. You don't see it growing around locally like the feral Lent Lily (N. pseudonarcissus) types, which are common.

Sounds awfully like the old Pheasant's eye - n.poeticus var recurva (which has more green on the trumpet than poeticus Actaea).

When is it's bloom cycle? Is it single headed? Right now, my triandrus narcissus are flowering, soon to be followed by the tazettas while the poeticus (I have 2 types) is usually latest of all.