This picture illustrates something else: sometimes the partial veil's point of attachment to the stalk is at its very base, instead of high up. In this case, the entire stalk (below the point of the ring) ends up covered with pieces of partial veil, and the ring ends up pointing up from the stalk (like a loose sock), rather than hanging down from it. The stalk in this case is said to be caligate, or booted, and the ring used to be called an armilla, rather than an annulus, and I am reviving the term since it comes in handy sometimes: for instance, many mushrooms now in the genusTricholoma used to be in a genus typified by this character and named after it: Armillaria. Why they're not there any more is another story, but in any case the armilla rather than an annulus is a useful way to help distinguish them from other white-spored genera with a partial veil, like the Lepiotaceae.
An extremely fine, fragile veil with a cobwebby texture is called a cortina. In the specimen at the center of this picture, a few strands of it can be seen extending from the stalk to the left of the cap, and broken strands are visible as a fluffy brown mass on the stem. A cortina is one of the major features of the genusCortinarius, and the threads of the cortina often catch its brown spores as they drop, making them visible as fine brown streaks along the stem.
A partial veil may also consist of a layer of slime. In this case, the slime usually coats the stalk after the mushroom has opened. I don't know of any special term for this and just call it a slimy partial veil in this database.