There's some science on training those muscles, even without upside down drinking. "Dry swallowing" while on an incline seems to do the trick just fine.
I suck in a lungful of air, hold it in as long as I can, then when I can't hold it any more, suck in even more air. Basically take in as much air and hold as long as you can, and the extra gulps of air I think open up and "reset" your throat.
Sometimes it takes a couple of goes but it has always worked for me to get rid of hiccups.
https://pmc.ncbi.nlm.nih.gov/articles/PMC9550520/