The situation was described as going onto a public network (give or take registration/authorisation, if not entirely free(as in speech)-to-use).
I'd be tempted to shell-script something to pick up the IP (
e.g., though that may suffer from both-ways NAT, when the truely local RFC1918 address would be a better direct link) and POST that info to a web-resource you control. The device that wishes to discover what to connect to just queries the external site, reads the "last known location" info and then tries a suitble handshake with that to verify it.
I wouldn't rely on hard-coding any RFC1918 address, TBH. Administrators (knowingly, or just by using the given auto-setup script for the hotspot service) might do all
kinds of weird things with the network masks and allowable ranges. You might be able to tinker with successive attempts (or a series of attempts to 'register' as one of several preconceptions until it doesn't seem to bug out) to see what works on a given AP that you can get away with.
(Somewhat the reverse, but with some similarities, twenty+ years ago I had my University account run on login (if detected as on a suitably static workstation) a check of what IP I was logging on from, see if it had a location (room, position in room) recorded in a simple text file and prompt me either to confirm (or note as unreliable) or else
provide (if not yet listed) the relevant location info. Then I could use this same personal lookup file to augment a "where my friends
1 are" display, plus a "who else (known or otherwise) is logged in in this room" one. A nice little toy that saved time and did my 1337-h4ck3r cred no harm at all...
)
1 I had some. I didn't always know their real names, without fingering their accounts, but they either existed or were very good at Turing Tests.