I think what you described with the tapered plug is probably in the jam chuck category. There are some variations on a jam chuck, one is you jam the bowl interior against something held on the spindle end and bring up the tailstock. Do the turning and then take off the little remaining stub with it removed from the lathe. Another variation is to carefully make a recess in some wood held by a faceplate or chuck. This recess allows the bowl rim to be tightly jammed into it. Carefully done, this will hold the bowl and allow complete access to the bottom but the tailstock should be used until the very last little bit of turning is done.
My favorite mechanical method, it to make what is called a donut chuck - lots shown online. This is basically a plate with 3 or more bolts around it (carriage bolts are handy) and a matching plate with a suitable size hole in the center. The bowl is sandwiched between the 2 plates and the bottom protrudes through the hole. By making just the "hole" plate you can accommodate various sizes of bowls and the hole does not have to be carefully matched. It is also a very secure method of holding the bowl - make sure the protruding ends of the bolts face the headstock!
My real favorite is vacuum chucking but that requires some $$ and setup.