You have lots of time for the first mission, up until 6:00 am day 2.
You can get the SMG in Paradise Plaza. It's on the blue awning above the photography shop. To get there, go up the first staircase on your right when you enter the plaza from the warehouse. You should be able to jump onto the blue bit about halfway up the stairs.
You can also rush to the gunshop at the start of the game (check you map, it's in the North Plaza), and pick up the Snipers and Shotguns there. Alternatively, one of the survivors who gets owned at the start of the game has a shotgun, and you can pick it up from him.