This is the same mayonnaise that comes in glass jars. It's an excellent thick mayonnaise, but it's too thick to work in a squeezable container The mayo should be thinner, like ketchup. It's not a refrigeration problem. It's the same problem at room temperature.
The problem is that the mayo doesn't sink to the bottom of the container, so it cannot be squirted out. When the container is full, you can give it a good shake-down but, when it's less than full, that doesn't work. The mayonnaise stays stubbornly stuck up the sides. No amount of shaking, bashing and squeezing makes any difference. You just puff out air, and the occasional flatulent-sounding splatter of mayo.
To get the final 30% out of the container, we cut it in half with scissors and scooped out the mayo out with a spoon, decanting into a normal Hellman's jar.
So, great mayonnaise, wrong container.