FFC. Possibly multiple FFCs, because AFAIK, >256 position FFC connectors do not exist (after 80, things start to get expensive). This has the added benefit that they can be connected to the circuit from different sides, possibly allowing layout optimizations compared to all 256 signals coming to a single point.
OTOH, if you need to put >256 signals through connectors to another subassembly, you most likely should think about building it in a different way, with increased integration so that more happens at fewer places. If there is some specific requirement which prevents this (such as abnormally high temperature where the signals are needed), it most likely also limits the choice of connectors, as well.
Flex circuits can be soldered in permanently, as well. Not sure what kind of equipment is used for this in industry, though. Tiny amount of solder paste and reflowing with hot air should work as long as you have a means to clamp the connections down. Some Kapton tape could work for this in small scale.