This first step is actually one week too early. This week should show how big an increase of infections will there be (or was there, depending on your time frame) after the first phase of reopening, started on May 4th.
Those data should have been evaluated BEFORE finalizing the reopening timetable; but they haven't. Everybody's hope is they'll be good enough to justify the possibility of this new phase, but what if they aren't?
In two weeks, so on June the 3rd, the effects of the new easing measures starting from Monday will start becoming evident. But an even bigger reopening is set for that date. Once again, the reopening schedule will be one week earlier than ideal to be able to define the new measures by fully assessing the earlier ones.
But, economy was pressing, and people are not good at thinking rationally AND more than two weeks in advance when they are scared of losing their job and money and are wary of the limitations.
Virus exploit human nature, and they are very good at it. Let's hope this bet (albeit at least it was reasoned) Italy took will pay off.