Most people who make this trip are finishing a stay on Amorgos, the rugged south-eastern island of cliff paths and deep blue coves, and heading for Syros, the capital of the Cyclades, with its neoclassical port town of Ermoupoli and a year-round city life. Some are catching a flight or a Piraeus ferry from Syros; others want a few urban days after the quiet of Amorgos.
The honest answer about this route is that the direct link hardly exists. Blue Star Ferries is the only company Ferryhopper lists, and its schedule cards show sailings from Katapola to Syros only in April and from Aegiali to Syros only in March, each once a week. The monthly table, which is based on last year's sailings, shows no data for any other month. The opposite direction, Syros to Amorgos, is much better served, so do not assume a return exists just because you sailed out that way.
That changes how you plan. Unless your dates fall on one of those spring sailings, treat this as a two-leg trip. Naxos is the obvious place to change: Amorgos has boats to Naxos on most days, and Naxos has sailings on to Syros. Paros and Mykonos also connect both islands in season. A night on Naxos or Paros can turn the change into part of the holiday.
Whichever way you go, work out your Amorgos port first. Katapola, on the west side, has most of the island's sailings; Aegiali, in the north, has fewer. Check which port your first boat leaves from before you book a room for your last night.