Train from Siena to Naples

I think it will be easier to go back to Florence. Either by bus or train. Both about 1 hours 30 minutes. The bus station in Siena is located in up town. The train station is outside the town. If you take the bus in town to the train station, it is about 6 to 10 minutes. If you don't want to go back to Florence but go direct to Rome and than Naples. You have to change train in Chiusi or Rome. Depend on what kind of train of you schedule. Your might have to change train couple of time. From Florence, you can go direct to Naples. The ES will take about 3 hours. By the way, the bus station in Florence is very near to the train station.
Have a good trip!

Is there even a direct train from Siena to elsewhere in Italy other than Florence?

The bus station in Siena is right in town. Take a bus to Rome. You'll end up at Tiburtina Bus Station which is right across the street from Tiburtina Train Station. Just a few stops to Termini and then you can get on the train to Naples.

Yes, it will be cheaper to take the bus to Rome and than train to Naples. But it will take a little more time. May be one hour to one and half hour.