Hello Sue and welcome.
I'm pleased to hear that you are enjoying the site and I hope it's helping you plan your holiday.
October is indeed a good time to visit. It's a month that was once described to me by a Greek guy as "sweet" and I think that sums it up nicely. The island is starting to wind down for the end of the season but everything will still be open and the boats to the beaches will still be operating. There will be fewer tourists around although Symi never really gets crowded except perhaps in August and you'll find that the locals will have more time to chat with you.
The weather is likely to be "sweet" too. The searing heat of high summer will have passed but it should still be nice and warm, say somewhere in the mid 20s. It is likely to be cooler in the evenings though so pack a cardigan or thin jacket to wear when you're out and about in the evenings. There is a chance you will see rain but it's very unlikely and will probably pass quickly leaving a lovely fresh feel to the place. I know you're not bothered about swimming but the sea will be very warm in October (it's had all summer to warm up!) so if you feel the urge to dip your toes in I think you'll enjoy it.
The weather is likely to be warmer at the end of September if you go for that option. The island won't be much busier than early October, just a little warmer.
You have probably noticed from some of the posts here that the ferry timetables change from year to year and the changes are rarely for the better. This year the ferry timings between Symi and the neighbouring islands are worse than in previous years so I'd urge you to check Andy's combined timetables posted in this forum before finalising your plans.
I'm impressed that you are planning an independent solo holiday so far from home, it really is a long trek from Australia!
Including Symi in your trip is a good idea in my (biassed
) opinion. Symi is a very safe welcoming place for solo travellers. There will be plenty of people, residents and tourists, who will be more than happy to chat and spend time with you if that's what you want or they will leave you alone to sit and read if you prefer.
Enjoy your holiday and please let us know how you get on.