What type of caulk/puddy/glue should I use to reattached one single 1" x 1" tile? The grout between it and the surrounding tiles is still 100% in place. I have the piece that fell off so no concerns finding a match.

The tile fell off the dam wall that is between the hot tub and the pool.

Water flows over the wall when the pumps are running. It happened a couple of years ago, possibly following snowmaggedon. Fortunately it didn't cause a chain reaction.

edit: the location is dry when the pump isn't running so I don't need to mess with lowering the water level to fix.

Never done this but know well that the subsurface is suspect at fault so
chisel out bad workmanship then consider these "glues" after more experienced weigh in w/advice

https://www.uline.com/Product/Detail/S-24796/Adhesives-Glue-Epoxy/Loctite-Construction-Adhesive-10-oz?pricode=WC3271&gadtype=pla&id=S-24796&gad_source=1&gclid=CjwKCAjw7s20BhBFEiwABVIMrcR0-zsssrJTA6cGnNfRrUMw5uIQJijR3g9wElxwJvzq_X6mrWnfCBoCesIQAvD_BwE

https://www.amazon.com/Pool-Patch-PTSRKW3-Setting-3-Pound/dp/B00Q2L3PZI/ref=asc_df_B00Q2L3PZI/?tag=hyprod-20&linkCode=df0&hvadid=693712983328&hvpos=&hvnetw=g&hvrand=15480646822808783074&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=9027906&hvtargid=pla-308903822808&psc=1&mcid=414334a72ea63a9aaaf23fdf7fb2c820&gad_source=1

https://www.amazon.com/Red-Devil-0497-Adhesive-Squeeze/dp/B000BQO2UC/ref=sr_1_4?crid=1J7M2DTT6XHWY&dib=eyJ2IjoiMSJ9.LLByeggQS7qr5GB-83JUBElp33AJ3F4K0fODcLb0O9Kj3xMRJTshBKtMoUp2NiKedoddFXHmrdkRy-Ykq3XXkklK18tJi9yvO0O_iKIjKcAYUike0qWw46N1kpxQnvUMkyhthLL2w7I3kWzsGaNQnS1Lb38KeeBWNyM7P2XszbsLS5iEv44RzqEL_yfHbPVKQDqZhBDmqMGS8Y6R1194RbJRG9sKViwYaWLaNDvzQNE.ONXhyi4qyKLOhUIH7TiUgzI5t6nkOxSIegFvKEevTFg&dib_tag=se&keywords=tile%2Bglue&qid=1720990725&sprefix=tile%2Bglue%2Caps%2C130&sr=8-4&th=1

Interested if you've found a good fix as I'm in a similar situation. One full tile and a smaller partial tile or accent tile have come off right by my spa spillover into the pool.

I'm guessing I can find some sort of adhesive/epoxy/whatever like tgivaughn suggested to get the tiles back in place even underwater if I wanted but I'm guessing I am going to have to lower the water level to touch up the grout between the tiles. I'm not sure how to go about matching the 'aged' grout but I'll figure something out.

Clean it real good and chip out some of the old grout if you can. Use a similar colored epoxy grout. You might be able to buy a small tub of laticrete or mapei from floor & decor

Thank you all. My wife beat me to the punch and ordered some type of epoxy. Once I get around to applying this weekend I'll follow-up w/near-term results and what she grabbed.

If something annoys her she's usually great at researching so she likely got something appropriate to fix it.