Could one use jittering to overcome non-positivity in propensity score estimation, i.e. complete separation in logistic regression?

I stumbled on "jittering" when struggling with logistic treatment model results (to estimate propensity scores) that ended quickly in complete separation with very few variables. They yielded propensity scores that almost failed to discriminate between treatment group and control group. The included variables (only some of them categorical), predicted "too well" the grouping (although the study units had been group-randomized to intervention versus none, before the baseline data collection that preceded intervention start).

I found that jittering has been proposed for discontinuity correction, smoothing and better confidence intervals. My idea would be to add random bits to candidate independent variables likely to predict treatment assignment, so as to weaken their relationship with treatment assignment and thus avoid separation; perhaps multiply, like when using multiple imputation for missing data. I searched but found little or nothing in the literature for this kind of use of jittering.

BTW, I found very relevant your argument for context-sensitive jittering (limits, shapes etc.).

Thanks in advance, even for a very spontaneous / ad hoc guess on this question.