Invertible Hasse-Witt for non-ordinary curves

Assume that $S$ is a smooth curve of over a field $k$ of characteristic $p>0$ and $f\colon X\to S$ is a relative curve over $S$ (i.e., the fibers are curves). It is well-known that when all the fibers are ordinary curves, the Hasse-Witt map $h\colon F^{*}(R^{n}f_{*}(O_{X}))\to R^{n}f_{*}(O_{X})$ is an isomorphism, where $F$ is the Frobenius morphism of $S$ (this result is due to Katz, Slope filtrations of F-crystals, in Asterisque '63).

Now, are there cases where the fibers are not ordinary curves but Hasse-Witt map is an isomorphism? Generally, are there reasonable conditions (of course, I mean besides ordinariness!) under which the Hasse-Witt map becomes an isomorphism?

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I presume $n=1$. Do you want examples in higher dimensions? Then there is a general definition of "ordinariness" for varieties over a field of characteristic $p$. Except for the local constancy of the direct images in families, it is pretty much related to the condition you require. See, for example, Part III of my Cohomologie cristalline, un survol, Expositiones Math. (1998). – ACL Dec 14 '11 at 7:05