Here are, in order, the most frequented islands of the Cyclades, so theoretically the ones you want to avoid:
Santorini, Mykonos, Naxos, Paros, Milos.
In fact, if there is an airport, there are more people.
Santorini and Mykonos also receiving cruise ship visitors are the big winners in terms of crowds
That being said, before June 20 you won't have too many people in Naxos, Paros and Milos.
Sifnos is not a sprawling island, you must confuse it with another island (and I wonder which one!)
If the hiking trails in Sifnos are excellent, Sifnos is also an island that can suit you even if you don't like to walk. It has everything that can make the typical postcard of the Cyclades with more authenticity compared to Santorini for example.
It is also renowned for its cuisine (and its pottery).
The same goes for Tinos which is even quieter than Sifnos but has fewer beaches, less known cuisine and almost no pottery but is full of beautiful typical villages.
Syros is the capital of the Cyclades and has its charm, however it is not typically Cycladic in style, no white houses and blue domed churches.
Know that having a rental car is also a parameter to take into account. Some islands are more or less well served by local buses.
Naxos, Paros and Sifnos have a good bus network. That of Tinos and Milos is much less efficient (especially in June)
The Naxos-Paros-Sifnos trio is also the one with the most beautiful sandy beaches.
Do not plan anything without having seen the ferry possibilities.
To know them see on ferryhopper
https://www.ferryhopper.com/en/
or on openseas.gr
http://www.openseas.gr/
(same information but with different presentations)
And if you are looking above all for calm and authenticity, see also Serifos, Kimolos, Folegandros and Amorgos