How to secure the bottom of a moving box?
The strength of an unmodified box comes from the folds of the flaps at the bottom (or top) and creates a long lever arm of sorts that is secured at the point where the tape is applied. If you trust the material from which the box is constructed, your efforts to secure the flaps properly will prevent a drop-out of the contents.

If you do not trust the material sufficiently, you can reinforce those folded points with corner material. I recently received a shipment of long slender metal rods. Inside the box, the rods were secured to heavy weight card-stock angles, similar to aluminum or steel flange-type material.

Of course, if you don't have easy access to such material and yet do not trust the existing material, reinforcement of those folded points with load-spreading design is your best bet. 

I've applied hot glue angles for the same purpose, extending the angles up the sides about three inches and into the box floor the same amount. Liberal application of hot glue provides for reassuring stiffness. In my more recent case, the shipment was a machine tool base of 75 pounds of rectangular steel. 

Your previous method of using foam sheet would tend to distribute the load more evenly than without, presuming that you do not have point-force items in the box.

A table leg will bypass all of the above suggestions, causing higher forces on a small segment of the box. Sometimes it's not how you close it up as how well the inside forces are spread around.

Good quality packing tape, perhaps a 3M or Gorilla brand product is suggested. Some of my tape purchases barely made it out the door before it started peeling away!