My best guess would be they went into major shock from the climate change of going out early and perhaps perhaps were hit even harder if they weren't fully hardened off fully first. There's nothing I can see from the photo that would say you definitely have leaf spot. Spotty necrosis can result from other things and even show what appears to be an active border as you'd expect on leaf spot.
If it is leaf spot, it's going to travel by contact. That would mean things that touch the plant and infection falling to the dirt and splashing back up in rain or with watering. Since yours are in containers you can isolate well by simply separating those containers. I suppose insect vectors may still be possible, which increased separation distance might help with. I just don't hear about insect vectors as much with the leaf spots as I do more general contact transmission. Also I'm not convinced you have a leaf spot problem.
Even in a best case scenario the plants will be set back by the shock and likely won't recover quickly. I'd keep them warm as much as feasible. If they're going to get cool, keep them out of the rain. Rain and cold is a bad combination. Even though they appear dry now, they look to me like they may have been too wet at some time recently. Control your variables and focus on giving them good conditions otherwise. Good watering and feeding practices as discussed above. Look for signs of the new growth coming in green. Don't expect the old growth to do anything other than continue to look bad or get worse.
That's my $0.02. Not a season-ender, but your plants will always be behind where you would have been if you'd waited longer to put them out.
Good luck!