I personally find my dexcom can read a bit low for the first 12 hours, so was getting fake night time hypo alerts when I put one on in the evening....
I just went without overnight to get back to a morning start up, but of you don't want to do that you can, as
@therower said, you can end the old sensor early (stop sensor is at the very bottom of the settings menu). Not sure how much help putting the new sensor on early is though, as you still have to wait two hours after activation for warm up, though I guess it might reduce new sensor inaccuracies?