what's likely happening is due to an increased fluid capacity by adding your cooler (and whatever else), you're at some point draining your reservoir after you start the engine which will allow air into the system. you can check this by starting the truck and have someone turn the wheel while you look down into the reservoir to see if you're sucking it down. Once you aerate the system, it will want to foam up you'll overflow the res once you turn the engine back off.

...what to do? ...get a bigger reservoir.