How can I automatically advance to the next member of a filtered group upon webform submission?
This is no quick solution (though I suspect there isn't a reliable one), but just as a note for future development--this situation of grabbing the "next call" is precisely why reserving contacts in the phonebank feature exists.  I think the real solution (that may be out of your immediate budget and timeframe Joe, but is what would actually tackle the issue) is to have a nice frontend on the phonebank.

Just as I call exposing a profile in create mode for semi-privileged users "intern mode"--for allowing someone to enter contacts without actually giving them full CiviCRM access--a phonebank listing would let someone go down the list of contacts reserved for them and potentially reserve a few more as the caller gets to the end of the list.  That would solve this problem in a non-Drupal-centric way, use the nice "reserve" feature to handle the situation in Chris's comment, and provide a complete experience for those running and staffing the phone bank.