Moving to a state with property tax exemption. When will it reflect on my mortgage?

Hi all, 100% P&T here and am moving to a state that offers property tax exemption.

I’ve looked on the DMV site and it looks like they need drivers license to match future address. Let’s say I close March 1st and 1st mortgage payment is due April 1st. What’s the quickest and most efficient way I can do this so that the tax exemption reflects immediately starting on the first mortgage payment?

Have any of you experienced a similar situation? How’d you do it?