Near Segesta Archeological Park. Navigate to Parcheggio Terme Libere Segesta (37.972849, 12.893917), which is located down a steep dirt path starting near the private Terme Segestane resort. Park in the parking lot and change into swimwear and sandals (some wading required; sandals with straps preferred compared to flip-flops). Bring a towel as well as water.
There are two ways to get to the hot springs, neither of which are particularly accessible. The easier (but slightly longer) version is: at the end of the parking lot, take the path into the woods. If there has been recent rain the ground may be very muddy with deep puddles.
A stream will appear on the left side. Cross the stream, being careful not to trip on the rocks. The water level will vary depending on rain but when I went it came up to around halfway to my knees.
You will then arrive at a fork. Take the right side.
The hot spring will appear on your right (you’ll be able to smell the sulfur).
In case the first path is not accessible for some reason, you can try the second path. Carefully walk down the steep hill at the beginning of the parking lot, which leads into a deep pond. Then scramble up the other side, and you will come out of the left side of the fork pictured above: continue straight to arrive at the hot springs. Below is a picture of the deep pond taken from the fork.
When leaving, drive back the same way you entered. Do not continue the other way from the parking lot (towards 37.973254, 12.896022), which is an extremely muddy path that leads to a private farm and not back to the highway, as Google Maps may otherwise suggest.
Bring swimwear and water. If you go at a popular time such as on the weekend or if it has rained recently, you may need to bring your own shovel to dig a hole (otherwise you can use a hole someone else has dug).
There are a few entrances to these hot springs. When I went the only accessible one was as follows. Drive to the hot springs entrance (22.89005758549187, 121.07007955941333). If you're driving from Luye City, it is located past the red Hongye Bridge but before the private Vakangan resort. The entrance is to the right of a traffic mirror and is currently boarded up with metal and warnings not to enter (in Mandarin). Park your vehicle, ignore the warnings, and enter from the left side of the metal boards.
Walk down the stairs until you get to a rock cliff. If there are other people at the hot springs they should be visible from here. You will see a rope you can use to rappel down the (~15 foot) steep face down to the riverbed. There are a few footholds but it’s not the easiest climb down. The warnings mentioned above say that the rope may be unstable; descend at your own risk.
Walk forward until you arrive at the riverbed. You should see many holes dug by others; you can use them or dig your own if you brought a shovel. Apparently most of the holes are dug by a 90 year old man.
To cool down, you can carefully hop into the river for a bit. Alternatively, you can build temperature and depth control for your pool by digging narrow conduits from the upstream river into your pool and from your pool into the downstream river. By opening the upstream conduit, river water flows into and cools your pool. The downstream conduit allows you to control the depth of your pool.
There is another (perhaps previously accessible) entrance next to the bridge. It did not seem accessible when I visited (too steep, path not clear), but this could change.