I don't see any need to buy two train tickets. If you miss the connection, you have four hours to get the second train (regional trains, which this surely is on to Siena). However, checking the Trenitalia site, I see that some of the connections from Chiusi-Chianciano Terme to Siena are in fact buses - run by Trenitalia? Can we assume you'd be able to catch the next bus from Chiusi-Chianciano Terme if you miss the train? I'm honestly not sure, but I'd guess so; the price is the same on the website, and you can buy it online from them.
And they don't run (trains or buses) constantly from Chiusi-Chianciano Terme to Siena. You might need to wait 90-120 minutes for the next one. Given that, I think I would try to time my connection at Chiusi-Chianciano Terme so the next train is an hour later, not more, so you won't have to wait so long if you miss it the original one.
However, I'd guess your chances of making the connection are good - your connecting train may even be on the same or adjacent platform, so you might not have to go very far to connect. Some of my tight train connections in Italy last year were like that. FYI, I found having the Trenit app on my phone very helpful; I could find out the platform of the next train ahead of time and not have to stress when I got off to try to figure that out in a rush. Trenit also shows train delays, and I found that some of my connecting trains were in fact delayed too, so I could relax a bit when changing trains, knowing the next one was a little late.