"secluded, smaller, less travelled and less touristy islands"
which can easily be combined with Rhodos:
Symi: daily catamarans from Rhodos
Tilos: catamaran twice weekly (usually Tuesday and Thursday) plus 2 slow ferries (actually Bluestar Paros)
Nisyros: catamaran twice weekly (usually Tuesday adn Thursday) plus 2 slow ferries (actually Bluestar Paros)
Chalki: catamaran twice weekly (usually Tuesday and Thursday); slow ferry once to twice weekly, according to the season; daily small ferry from the port of Kamiros Skala, reachable by bus from Rhodos city.
Have a look at https://www.greeka.com/dodecanese/

"secluded, smaller, less travelled and less touristy islands"
which can easily be combined with Santorini:
Anafi: 3-4 slow ferries weekly, all in the middle of the night
Sikinos: several boats and catamarans, almost daily in high season
Kimolos: several boats and catamarans, almost daily in high season
Sifnos: several boats and catamarans, almost daily in high season
Amorgos: several boats and catamarans, almost daily in high season
Schinoussa: can be reached about 3 times weekly from either Amorgos or Naxos
Iraklia: can be reached about 3 times weekly from either Amorgos or Naxos
Have a look at https://www.greeka.com/cyclades/

Those islands mentioned by neckervd in the last post are all smaller, but I'm not sure they will all be less crowded in July. Symi, for example, gets lots of day trippers from Rhodes, and Nisyros lots of day trippers from Kos. Small islands have less accommodation and tavernas, so even if there are fewer tourists they can still seem crowded. The ones least likely to be crowded are the ones hardest to reach (no daily ferries).

The excursion boats to Symi (from both, Rhodos and Datca) are not really big. People coming with them may form crowds in the monastery and in the main city, but not at places where people who stay on the Island will spend the time between 10am and 4pm.
Idem for Nisyos: day trippers change usually from excursion boats to buses in order to see the volcano, may then have dinner somewhere around the volcano, spend about 2 hrs (or even less) at Mandraki and leave the Island again.

Chalki is easy to reach but had almost no day trippers when I was there in July 2016.

Hello,

Regarding the less touristy islands:

Kythnos, Sikinos, Kimolos, Amorgos in the Cyclades
Donousa, Irakleia, Koufonisi, Schinoussa in the Small Cyclades (Island hopping there would be nice, especially if you are staying in Naxos)
Arkioi , Chalki, Tilos if you are planning to head towards the Dodecanese.
Gavdos, near Crete
Antipaxoi, in the Ionian Sea (generally the Ionian is better if you have plenty of days to go sailing)