How to repair a basement ceiling's old school interlocking tiles?
I think you may need a whole bunch of new ones, or a completely different ceiling system (as you prefer). Those should be attached to the ceiling structure, and the backside of the fallen one looks like parts that should support it are ripped away.

It's been a long time since I have used any of these - I vaguely recall that they would latch into others on two sides, covering the fasteners (staples, usually) of the ones they locked into, get stapled on the exposed sides, and have the next tile cover up their staples. So your expectation of having to work to a wall sounds about right.

A grid-type suspended ceiling system is similar, but gives better access above it if needed as the panels just sit in the frame; though it takes up a bit more headroom (which you might not want in a basement.) You could drywall and have similar height (it may be "louder", though), or you can find this type of acoustic tiles (but you're probably going to have to replace them all, for color &amp; pattern match if nothing else. 30-40 years of aging shows when you put a new one right next to it.)