Southern Arizona just got a massive amount of rain in the past week, and even before that I was seeing a large amount of water coming over Winn Falls (which is on the Cima Creek drainage) about two weeks ago. I suspect there's water flowing even as high as Cima Cabin, and if not it'll be close by. The upper spring above Cima Cabin can be hard to locate but had good water under the very heavy lid in late May.
I suspect all the springs are and nearly every creek is flowing right now. I was just working in the Huachucas over the past week and there was water just bubbling up out of the ground on ridgetops at 8,500' all over there.
[ Anita Spring ] ,
[ Eagle Spring ] ,
[ Tub Spring ] (surface flow, drinker is not connected), and
[ Booger Spring ] are all pretty reliable and I've never seen them dry thus far.
[ Headquarters Spring - Chiricahua ] used to be but hasn't been flowing into the box quite as well lately so can be hard to retrieve from, though the box likely has rainwater in it.