Most of the spring areas across Tuscany have been taken over by hotels and spa centers, and there are many hot springs in the region. Any town you see with "terme" in the title means it has springs, but you wouldn't know it other than maybe because there is one central establishment over the springs that control access to them. Since you're in Lucca, definitely Bagni di Lucca is the closest and spa treatments can be had at these centers and hotels. It is up to you to choose which "treatments" you want and decide from there.
All the centers work on day passes, since they aren't hotels. You just show up and basically gain access but you can also book special treatments.
As for the hotels, they also offer day passes, look for "day spa" as a sign that they have packages for spending a whole day at the spa without having to stay there.
For example, Hotel Bagni di Lucca has a whole pdf with its pricelist here:
https://www.termebagnidilucca.it/admin/img-multimedia/carta-trattamenti-11.pdf - the pdf is from 2018, so it is just to give you an idea of what normally is offered.
The thing is that, right now, they have a notice on their home page saying their pool (fed with the springs) is open (day pass costs €20) as well as the hotel but that the spa center is actually closed. I suppose they are waiting for the season to fully take off.... but it has, at least here in Florence, so maybe their site is just not yet updated???
I would suggest you contact them directly and ask whether the spa and treatments are once available. Fall is still many months from now, so it is likely everything will have reopened by then. So you can't go off from what is available right now.
This hotel is right next to some of the three main "thermal" centers: the Jean Varraud, Casa Boccella and Centro Ouida.
You can read more about the history of the springs and the various establishments on this website, in English (!!!), very interesting:
http://www.bagnidiluccaterme.info/en/argomento/Le-terme/Bagno-Caldo-o-Jean-Varraud.html
You should know that I found that only Bagno Bernabò is open as an establishment. Spa centers suffered long periods of closure during the pandemic, having been closed for almost the entire two years.... so many might not reopen at all. But I found news from this fall that this one was slated to reopen by December, which I hope it did.
Are you staying in Lucca at a hotel where you could email them and see if they can confirm whether the spa centers are open? Otherwise, I would suggest heading south instead to San Giuliano Terme, closer to Pisa.