is there a secondary voltage overshoot when switched on?
Non-saturating core scenario
The secondary voltage is proportional to the rate of change of primary current and, in the inrush situation, the initial rate of change of current is the same as when it settles down so no, there won't be an over-voltage situation on the secondary.
Saturating core scenario
Due to the DC offset in primary current when inrush occurs, the core can saturate. The effects of core saturation can mean large current spikes but, these are simultaneously met with significant reductions in the core permeability. In effect, the two "problems" cancel each other and, there is no secondary overshoot.