Oak, the Shebeest jersey has a zippered pocket and nothing would fall out of there. I have a running jersey just like it, without a pocket and I think both of them fit me exactly the same, so I would wear the cycling jersey for running.
I use a Nathan pack with a teeny pouch and I can put my phone, license, and a couple of shot blocks in there. It's kind of bulgy with all that, though.
I mostly run when it's cold out, so I have jacket pockets.
2015 Trek Silque SSL
Specialized Oura
2011 Guru Praemio
Specialized Oura
2017 Specialized Ariel Sport