If beaches are not a deal-breaker, then consider former Eastern European countries or the Balkans.
What are the most important criteria for selecting a place? The one aspect that stands out to me from your post is "beaches", which really narrows down the options in September/October. If you're looking to swim, then going to Southern Europe or Turkey makes more sense because the water will be warm. I don't think there is a best place for solo travelers because solo travel is what you make of it (irrespective of the country)...I think your specific interests should guide the choice (I can't think of any place that doesn't have great architecture, scenery, good food, and history...but not all will have beaches). If by solo travel you mean that you have to be more budget-conscious (because there's no one else to split the cost with), then definitely consider places like Portugal, Italy (including Sicily), Greece, and countries that are not on the Euro like Turkey. If beaches are not a deal-breaker, then consider former Eastern European countries or the Balkans.