Patmos is very close to the island of Leipsoi, Agathonisi, Marathi and Arkioi. Leros is also near and a daily cruise to this island is something that should not be missed. Leros used to be famous for the psychiatric hospital on the island but nowadays it is a destination with rich tradition and many beauties, even for one day!
Other larger islands nearby are Samos, Ikaria and Fournoi and Cycladia suggests that you should definitely dedicate some days of your stay to one of those as well.