You just triggered one old itch, so this is a partial (both meanings) answer only. For discrete-time, uniformly sampled STFT, one often uses the usual fixed-window with $3/4$, $1/2$, $1/4$ overlaps, and somehow uses direct closed form inverses. However, as long as there is redundancy (and completeness), there exists a infinite option for inverses.
In a collaborative work (with Jean-Christophe Pesquet and Jérôme Gauthier), following other works on vector-frames, we used the formalism of complex oversampled multirate filter banks
to model analysis STFT, with the aims of
- checking conditions under which the STFT were invertible, given a hop between time-frames,
- building "better" inverses (better localized in time or frequency) with optimization.
Our aim at that time was double:
- for data in higher dimensions $d$, reduce the STFT redundancy $r^d$ as far as we could,
-for 1D data, reduce the number of channels and increase the hop as we could as similar quality for real-time event detection.
All was based on polyphase matices, as we did consider windowed transform only as a special case; so the main outcomes were the following:
- given any redundant analysis filter bank (complex or not, windowed-DFT like or not), check whether it is invertible
- in most cases, anything redundant is almost surely inversible,
- windows with negative values can be used, liked Kaiser windows, see On the Optimization of Oversampled DFT Filter Banks, 2007
- you can reduce the degrees of freedom in the oversampled synthesis to perform better concentrated inverses after selection/denoising for (close-to) real-time computations,
- ensure the Hermitian character in the inverse for scalar threshold.
There were remaining issues, among which: the difficulty to go back to the orthogonal setting when the redundancy tends to one. Weirds behaviors were observed, possibly related to Farey sequences, but I have no definitive clue. And
References can be found in Optimization of Synthesis Oversampled Complex Filter Banks, 2009. Some related codes is available at SURE-LET Optimal oversampled Complex Filter Banks synthesis toolbox. We do plan a better release, but it has been remaining upcoming for months.
However, we did not investigate the non-COLA effect, and the necessary condition is not stated. My belief is that it is not necessary (due to redundancy), but this should be proven.