I have see the method PCHIP in matlab that implements the monotone Hermite interpolation method which was originally proposed by Carlson in 1980s. It seem to accomplish the goal of preventing the values go outside of the range. I have not seen the error estimates results but I guess it does as good as cubic polynomial and locally $O(h^4)$. Now, there are more recent ENO/WENO methods and their multiple "kids". Would like to hear why these methods stand out and why they are "better" or worse compare to monotone Hermite?
PCHIP is not a conservative reconstruction, making it inappropriate for conservation laws. Furthermore, hyperbolic problems have discontinuous solutions so there is generally no benefit to a continuous reconstruction. Conservative monotone spline reconstructions are being investigated by the UK Met Office for use in tracer advection for atmosphere modeling, see papers on the multi-dimensional case, quartic splines, and applications. These methods are relatively new and are not currently popular with many other groups. Some reasons for this include