Hello every one!
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It appears that the bus service is with a good frequency in Amorgos during July and August, either from Katapola or Aeigiali.
However, there are no regular buses to the south of the island.
My question is: Is the south of the island worth visiting to justify renting a car? Are there other places where having a car is necessary for visiting? If so, for how many days is it recommended?"

For Astypalea, given that the island is not very large, do you think it is sufficient to rent a car for just one day? Additionally, it would be practical to return it at the airport to take a flight.?"
Thank you for your help!

All depends on the places you plan to visit.
Buses cost almost nothing, but they don't go everywhere.

Katapola - Chora isn't a probelm, neither Katapola - Egilai and from Egiali into the small villages in the vicinity.
There are furthermore taxis at Katapola and at Egiali.

Livadia - Astipalea - Maltesana by bus isn't a problem in hgh season. But there ae tons of other interesting beaches that you can only reach in long hikes or by taxi.

I suppose you know that there are ferries from Egiali to Astypalea and from Astypalea to Kos about every second day, plus the once weekly direct ferries Kos - Katapola - Piraeus and Piraeus - Astyplea - Kos