First of all, the scheduled departure time is important.

If you miss a connection and the next train leaves after midnight you should speak to staff as soon as possible. They might give you a proof of delay and you won't enter the new train (otherwise it would trigger a new pass day).

If the next train leaves before midnight, log the new train on Rail Planner as usual. Delete the old one if you had already added it.

In any case don't forget to claim compensation: 12€ for 1h delay, 24€ for 2h and more.

Btw I'd tend to not rely on the last connection of the day, it could be risky in some countries. Feel free to post your connection here and we can advise. :)

Thank you for your response.

I bought a global pass and a plane ticket to travel to Europe next February,and I'm currently studying about the Eurail pass.
For example, 

I take the train from Milano departs at 06:10 and arrives in  Rapallo at 08:17.(First reservation)
After a short look around Rapallo, 

same day, take the train that from Rapallo Station at 17:08 and arrives at La Spezia Station .and then  get off at La Spezia Station at 18:45,(Second reservation)
After spending time eating and drinking coffee, 

on the same day  I also made a reservation to take the sleeper train leaving at 23:53 and arrive in Salerno at 09:11 the next day.(Third reservation)
However, if the train that I am supposed to take at La Spezia Station a t23:53 is delayed and

I board after midnight, (For example, if boarding the train after 00:01)
then what happens to the date of global pass use?

I'm realy sorry I'm not good at English

It doesn’t matter if the 23:53 train is delayed. Only the scheduled departure time matters.