Starting at the point the inside opened floor now is, pull up and form the cylindrical form to the thickness that you feel is appropriate for the given form you asre making. And generally shape this form so that the part now near your eyes is the way you want the BOTTOM to look. (Yeah... the form is currently sitting upside down!)
Cut the piece off the wheelhead and set this aside so that the (current) TOP stiffens to leatherhard-ish. Depending on the weather conditions you might have to wrap the lower thick section with some plastic to keep THAT part from drying too much. But because it is thick and solid and base-down.... it will dry WAY slower than the hollow top part. Let it stiffen up til the top will well support the weight of the bottom. No more than that.
Then, using some clay for a nesting chuck to support and adhere the partially finished piece (not a Giffin Grip! ), invert the partially dry form back on the wheel in the opposite position to the way it was originally thrown (what was up is now down).
Like pretty much normal, open the thick still wet clay at the top through into the open area of the lower form. Pull up the walls from the juncture with the stiffer clay. By repeatedly wetting the area where the transition from leather hard and wetter clay happens you can re-soften this area a tad and facilitate some continued shaping at the juncture.
A bit of practice with this and you'll wonder why you did not think of it.
^^ Wow... consider my mind officially blown! I'm marginally terrified to attempt what you're describing... but I'm going to do it anyway. Thank you.
Cass- all of these are for a project (in brief- I'm making indigenous instruments with surface decorations of endangered species from the same region)... so I'm trying to follow tradition in terms of how I'll mount the drumhead. These guys will get a woven rope attachment for the skin.
I'm actually not much of a drummer- I play octave mandolin, but leave the percussion to people who are more adroit than me. So far, most of these have a pretty sharp sound- almost like an Indian tabla.