Edit: There are two Rock Canyons, they are the last two before slide... I also mistyped Rock vs Rook
I have not done this personally, so take it with a grain of salt, but my research into the area suggests your best odds in Snake Gulch are the are Springs along Snake gulch at Wildband, Willow, and Rock(not Rook), and that the Table Rock Canyon spring is not reliable.
There is a spring report near the intersection of Snake and slide, but it is unclear from reading triplogs how far up slide it is.
When you get near Ed Hatch cabin the pipe output from the spring there is broken, but perhaps you could still find the source.
The first water source I have marked below that in Kanab is around N 36d29m44s and W112d38m17.5s
Hike Arizona it is full of sharp, pointy, ankle-twisting, HAZmaster crushing
Hike Arizona it is full of sharp, pointy, shin-stabbing, skin-shredding
Hike Arizona it is full of striking, biting, stabbing, venomous