Are these islands realistic, ferry-wise?

Have you checked ferry schedules at www.GreekFerries.gr and www.DirectFerries.com?

While waiting for Nicksy's suggestions, if you want to add Kalymnos, this island hopping is doable with direct ferries:

1. From Patmos to Kalymnos: daily ferries
2. From Kalymnos to Astipalea (twice a week, Anek Lines)
3. From Astipalea to Amorgos (2 to 3 times a week - Blue Star Naxos)

It should be possible to include Fourni with a Dodecanisos Seaways ferry, but I'm not sure what the most efficient order would be.

Check the September 2025 timetables on openseas.gr

Since Astypalea and Kalymnos have airports connected to Athens, I would also look into how it's possible to start/end the itinerary with a flight to/from these islands.

We really enjoyed Kalymnos, we spent six days there last spring. If you like climbing, you'll be spoiled for choice; if not, there are plenty of other things to do.

Good site to check ferry routes times is
Www.gtp.gr

The most logical order for those 4 islands would be Fourni-Patmos-Astypalaia-Amorgos or the other way round. Both ways you can include Kalymnos between Patmos and Astypalaia.

Ferries:
Fourni -> Patmos: Dodekanisos Seaways on Fridays and Sundays.
Patmos -> Astypalaia: Dodekanisos Seaways on Monday
If you would go the other way round it will be on Thursday from Astypalaia to Patmos and on Sunday and Thursday from Patmos to Fourni. The last Dodekanisos Seaways sailing from Astypalaia Northbound is on 24 September this year.

Astypalaia -> Amorgos (Aegiali): 3 times a week (Tuesday, Thursday & Friday) with Blue Star Ferries. The schedule beyond 31 March is not published yet but normally it should be like it has been for many years now.
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If you want to include Kalymnos you have Dodekanisos Seaways:
From Patmos to Kalymnos every day except on Thursday. There are 5 days where you have 2 sailings.
From Kalymnos to Patmos there is a daily sailing and on 3 days (Friday, Saturday, Sunday)
Next to that there are also 2 sailings weekly (in both directions) with Blue Star Ferries. But in one direction it's late evening and the other direction it's in the middle of the night. So these are probably not very convenient.

Just to add: if you plan on taking Dodekanisos Seaways for some of your trips, subscribe for their loyalty program through their website. As a member you get 20% discount on every ticket. Every person has to subscribe individually.
Dodekanisos Seaways itineraries 2026: https://www.12ne.gr/en/itineraries
Dodekanisos Seaways loyalty program: https://www.12ne.gr/en/bonusways

In the tender for the subsidised routes for the next 4 years I spotted a new route Piraeus-Astypalea-Kalymnos-Kos-Nisyros-Tilos-Rhodes. That would be once a week. It remains to be seen if a company is interested (or if multiple are) in that route and if so which company wins the bid.
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Besides Dodekanisos Seaways there are some others on these routes:
SAOS Ferries goes 3-4 times a week from Kalymnos to Patmos and also the other way round. However there is nothing known yet about the outcome of the tender for the subsidised routes for the next 4 years.
The same goes for the ANE Kalymnou ferry that normally goes 4 times/week on the Kalymnos-Astypalaia route.

Fyi: I will be on Astypalaia from 7 till 14 September and on Kalymnos from 15 September till 20 October.

I’m on Astypalea 31/08/26 - 10/09/26 and Kalymnos 10/09/26 - 14/09/26, before heading off to Leros. Looks like our paths will cross on Astypalea again this year Nicksy.

Dorothy, your trip sounds fabulous!

Yes, will definitely meet you in Livadi on 8 or 9 September. We will see you at the Seatrack there :-)

Wow, thank you so much!!

Nicksy, JoLui, Kenko!

Can’t thank you enough. Nicksy, you’re just amazing! The hard part of our September planning is done. Now just the fun part. Leaning toward omitting Fourni and adding Kalymnos since Kalymnos is workable. Two new islands, two old islands.

It would be really nice to have a meetup, Nicksy and Sueevie, in September. Will add it to the calendar in case it works out.

BTW was NEtrekker on ta. Nicksy, you’ve helped me many times.