Transitioning from quantum to classical random walks on the line
Great question. Actually, the same question popped up in something I was working on a few months ago (arXiv:1011.1217). It seems that any natural kind of decoherence leads to behaviour that looks initially balistic, but which becomes diffusive as time increases, so you are transitionioning between a $t$ regime and a $t^{\frac{1}{2}}$ regime. See figure 2 in the above paper for an example of this. This seems to be the natural behaviour as your state gradually loses coherence.

This would seem to suggest that the variance only ever scales as $t$ or $t^2$, and hence the walk spreads as $t^{\frac{1}{2}}$ or $t$.

However, exactly the same thing happens in quantum metrology when noise is introduced, but can be overcome to yield an intermediate scaling (see for example J. A. Jones et al, Science, 324, 5931 (2009), arXiv:1103.1219, arXiv:1101.2561, etc.). One way this can be achieved is by making intermediate measurements.

Imagine you measure the position of the walker after every period of time $T$ collapsing the wavefunction, and allow free evolution in between. Now, imagine we want to evolve the system for total time $t=nT$. Then the variance in the position of the walker after this time will be $\mbox{Var}(x(nT)) = \sum_{i = 1}^n \mbox{Var}(x(T)) = n \mbox{Var}(x(T))$. In the absence of other decoherence we know the walker moves ballistically, and hence $\mbox{Var}(x(T)) = T^2$, and so the $\mbox{Var}(x(t)) = n T^2$. However, as $t=nT$, we can take $n\propto t^k$ and $T\propto t^{1-k}$. Thus $\mbox{Var}(x(t)) = t^{2-k}$. This way you can achieve any intermediate scaling, by chosing the measurement interval appropriately.