Do I Need to Add Another Chimney?

looks good, could be a little taller, although it's an ouside chimney, with a 8/8 liner you should be fine (which if it's outside dimensions are roughly 12x12 it will have), especially if the water heater (oil fired??) vents into it.(water heaters tend to keep the chimney warmer because of their firing cycles and waste heat, thus giving a better overall draft more than compensating for the additional fluegas volume requirement)

It would be pretty easy to extend that chimney if need be, cheap too.

Presently, the oil burner is a tank-less design.