There is no reason for fitted propensity scores to have mean or median equal to 0.50 as there is not strict assumption that dictates such ranges.
At first instance, I think it will be more relevant to "clip" your existing PS slightly (i.e. drop instance outside [0.025,0.0975]) and use the PS as planned (through IPTW, inclusion as additional covariate, etc.). Lee et al. 2011. Weight Trimming and Propensity Score Weighting provides more detailed advice on this. Lee and his co-authors found that clipping is not always beneficial for certain kinds of estimators.
Particular to the issue you describe I would also note the following:
- if the range of the estimated PS is too narrow this might suggest that a potentially confounder is unaccounted for. Simply put, we do not have ignorability. The potential outcomes $Y$ are not independent of the treatment variable $A$ even if we condition on some other variables $X$.
- we should always aim to have common support between treatment and control groups; if we do not, there is a possibility that the differences we are observing in PS ranges is due to some form of selection bias.
- we might simply overfit our treatment assignment data $A$. This would have the direct consequence that our PS estimates are artificially over-determined. We should focus into getting a well-calibrated and properly estimated method to estimate the PS to begin with.