A reminder: the context is: A (married) woman may go out with a קלתא on her head and fulfill at least the biblical requirements for head covering (and in semi-private situations, e.g. a courtyard, it would even be enough rabbinically).
קלתא isn't a really familiar word. (Jastrow defines it as vase-shaped basket, or women's work-basket. My guess is that he's basing it on the Rashi we're about to look at, since none of the other texts he cites have any particular clues as to its shape, although they do all reference it as an item specifically used by women. If you want a look, it's on page 1383.)
On the top of Ketubot 72b, Rashi defines it as:
קלתא: סל שיש לו בית קבול להולמו בראשו ובית קבול מלמעלה לתת בו פלך ופשתן.
Kalta: a basket that has an indentation to attach closely to the head, and an indentation above to put a spindle and linen.
A קלתא is basically a hat that you can store things in. But this does seem to be some sort of starting place for the practice of wearing a hat with one's own hair showing. (Although I haven't looked at sources for that practice yet, so I don't know how it will actually be set up halakhically. This does seem to be where I'd start, at least, since it is at least sufficient in semi-private space.)
However, it's also an item that seems designed to be removed once you're setting yourself up somewhere. At least, I'd find it easier to take my spinning out from a basket at arm level, rather than one directly above my head. I wonder if that, more than it's incomplete coverage, might be a source of why it would be viewed as insufficient covering in public. It feels a lot like why I don't often wear hats on their own- hats often feel like things you wear outside to stay warm or keep the sun off, and then take off when you get indoors. It seems sort of tempting.
If that temptation to remove it were the reason that the קלתא is deemed insufficient in public- it might well open the door to hats that don't cover all the hair...
An irrelevant aside- I have a hard time picturing a basket that adheres closely or joins to the head.
What do you think?