Consider an object that floats in water but sinks in oil. When the object floats in water, most of it is submerged. If we slowly pour oil on top of the water so it completely covers the object, the object...
A) moves up.
B) stays in place.
C) moves down.
Solution: A. With the oil atop the water, there is an additional buoyant force on the object equal to the weight of the displaced oil. The effect of this additional force is to displace the object upward until it finds a new equilibrium location.
You might object that there is no way for the oil that is only on top and on the side of the object to exert an upward force. And you would be right. Using the formula for buoyant force (twice) would give a correct calculation of the combined force of the oil and water on the object. What really happens is that the pressure (and downward normal force) on the top of the object goes up. And the pressure (and upward normal force) on the bottom of the object goes up even more (the horizontal forces change but still cancel). A complex calculation of these forces would give the same net result as the calculations of buoyant forces. The buoyant force can be thought of as an accounting trick for multiple forces, not a new force.