Hi there! Love to see you here!

Night trains that are boarded before midnight count as the day of departure (even if delayed, as long as it was scheduled to depart before midnight CET/CEST). Thus, for Hamburg to Innsbruck you'd need a travel day, and for Innsbruck to Bologna you would need a separate travel day. As for in/out days, the same applies as above, so while you can use it from Florence all the way to Hamburg you cannot use it after leaving the night train. Your in/out day (if you are German) is therefore on the day you start in Florence.

Hope this helps!

For the night train back to Germany it all depends on whether the night train crosses the Austrian/German border before or after midnight if you can continue travel after leaving the night train.

If the night train crosses the border before midnight, your inbound travel day will be on the departure date of the night train. 

If the night train crosses the border after midnight,  you can get inbound travel day on the arrival date of the night train if you manually enter the night train journey split in 2 parts. One part from Innsbruck to a station in Austria that is passed after midnight, and a second part from that station to Hamburg.

Thank you so much @AnnaB ! See, on my travel back day I don’t need to go further than Hamburg because I live there, so by leaving the night train I’ll be home. Just looked it up and the train will cross the border before midnight. But do I have to leave the train then or is it okay if I check in in Innsbruck and stay until Hamburg? Sorry if I am a bit slow to get the hang of it

For this journey it might make more sense to simply book without an interrail pass. That’s what i would probably do, but it depends on your dates.

An adult 4-day interrail pass costs 283eur = 70.75eur per travel day.

Using 16 Oct as an example with prices from nightet.com for Innsbruck-Hamburg:
A seat on the nightjet will cost 9.90 with an interrail pass or 34.90 without.
A minicabin will cost 35.90 with an interrail pass or 79.90 without.
(On 17 Oct the minicabin costs 99eur.)

So, 70.75 for an interrail day, plus 34.90 for a minicabin = 105.65 for that stretch.
Or, 79.90 for the same minicabin with no interrail pass.
 

For Innsbruck-Bologna and Florence-Innsbruck it would also be much cheaper booked directly with Trenitalia for example (40 to 50eur per journey is a good idea of the price you could get, rather than 70.75 per journey + res fee for an interrail day).
 

Thanks so much for your reply, @finners :) But see, I already got an Interrail pass with the discount they offered not long ago and I have to travel on weekends so the Nightjet would be more expensive. But you’re absolutely right, I’ll always have to compare the costs before.

I’m still not sure about my last question, though: Can I board the train in Innsbruck back to Hamburg with my inbound day although it crosses the border before midnight? Would be so glad if anyone had an answer to that

The simple answer is 'yes'.

See 

https://www.interrail.eu/en/support/interested-in-interrailing/what-is-a-travel-day#:~:text=Night%20trains,the%20validity%20of%20your%20Pass).

My own experience:

In 2022 I took a day train Vienna-Berlin, then 3.5 hours later at about 9pm took a night train from Berlin to Malmö, arriving at 7am the next day. 

One pass day was needed for this.

If you get on the Innsbruck-Hamburg nighttrain on the same day as your Florence-Innsbruck train, then that is only one pass day.

So, your journey as explained will use 3 pass days:

Hamburg-Innsbruck -1 pass day

Innsbruck-Bologna - 1 pass day

Florence-Innsbruck & same day depart on Innsbruck -Hamburg 1 pass day

You could use the remaining pass day (if you have a 4-day pass) for Bologna-Florence or any other rail journey.

Thank you so much @finners , that’s REALLY helpful, this community is the best