@Sanjana I don't know who brought this following paper to myow attention, maybe it is you, maybe it was the bolb, but considering that you were so interested in the $\hslash$ expansion to get semi-classical physics with quantum corrections, there is a non-trivial version in
arxiv.org/abs/1009.2313 and at the one point where it referenced alternative schemes in ref number 7, there were 4 papers listed, the most relevant is
arxiv.org/abs/hep-th/0405239