Its a pretty generic Geneve automatic dial unfortunately. It appeared in dozens of references (note: MANY dials were available for each reference!), with no real rhyme or reason.
When going the OTHER way, you can search by dial diameter (that one looks pretty big?) and movement to try to find one that matches your case, but I'm unaware of any way to go this way. Noting of course that you have a tougher chore of figuring out one where the movement is valid too.
One idea might be to search the omega website here (
https://www.omegawatches.com/en-us/vintage-watches) for watches with a 565, THEN cross-reference years available down based on the movement serial number. BUT that would still get you a fairly sizable and incomplete list (as that database is awful for completeness). I WILL say that the dial with the extra spacer looks similar to a 165.002 I just had apart, but that is a no-date model.
That said, EVEN if you figure out which case it goes to, you'll have a quite difficult time finding a case. They don't get re-sold very often, and the opposite happens WAY more often (as gold cases, which hands/indexes says is a good possibility, are worth more as melt, so watches are melted and movement/dial/hand sets are sold off like you have).
SO you have a pretty up-hill battle figuring out which cases had that dial/movement combo, and an even larger up-hill battle finding a case for it.
