we have a soft-sided cooler that is slightly bigger than 6 pack size and it works great, it fits under our stroller which is only a small umbrella one. It's made by Arctic Zone, although I don't know if that makes much difference. We did have a cheapy one to start with, and that was hopeless as it dripped cold water on you once the ice started to melt. This one doesn't drip and in fact, the ice doesn't usually melt even on a hot Florida day.
I looked at getting a small hardsided cooler but there was no way it would go under the stroller.
Bev