You won't necessarily need a vest for the shorter runs, but I do recommend finding a way to carry your phone 'hands-free'.

There are various options such as arm straps or belts.... or even a pair of shorts with pockets

This Spibelt (LINK) is a good idea, it can also be used to carry other items if required such as money/cards/keys etc. They do come in different sizes, so it is worth checking it will fit your phone before ordering.

I don't know what shops you have near you, a local Sports/exercise shop should have some running accessories available