Models of computation strictly between classical and quantum in terms of query complexity
One easy way to come up with such a model is to first create a restricted model of quantum computation that can still do something non-classical, and then just give it classical computation for free.

An examples of this strategy is the one clean qubit model (along with a BPP machine). Some references: On the Power of One Bit of Quantum Information, Computation with Unitaries and One Pure Qubit and Estimating Jones polynomials is a complete problem for one clean qubit.

Another example would be to have a log-depth (or polylog depth) quantum circuit with access to a classical computer. This will yield something like $BPP^{BQNC}$.