There are day trains. Try to check the interrail planner for today, from Rome to Messina for example. 
On which date are you planning to travel? You can’t book reservations too far ahead. They open it later, when the final version of the timetables are published.

I’ve noticed these trains (in early June there are some too) as well, but these are missing for August. So I suspected I am too early.

Yes, you are correct. Try again later.

There is a timetable change in Italy in mid of June, so not all trains for summer are yet shown/available for booking. 

This Q has been posted-for It in general, about a few 100 times now. In IT they like it LATE=ritardi (they also like to strike very much=scioperta). And one can only get out of any computer what someone has before put in. Simply use samedayofweek for global planning and do not worry about full trains now. If that would really be the case, it would also make no sense to run a train network at all.

For Italy, always check on www.trenitalia.com.