I'd like to see a system where panhellenic makes up a list of all pnms released from recruitment after each round and distributes the list to the houses. Any house that wanted to give a released pnm a second chance could invite that pnm to the next party without counting her against release figures, and would be free to extend her a bid without counting her in quota. That way, houses are never forced to invite or bid any pnm they don't want, but they have an incentive to re-invite pnms who might have slipped through the cracks due to shyness, lack of connections, whatever.
The last time I suggested this, someone pointed out that sneaky pnms might collude with their #1 house and deliberately be rude at every house in order to be cut, with the understanding that #1 would invite them back as an "extra." I can see that as a risk in small systems, but would it really happen at a place like UT? Wouldn't actives in the other sororities talk, realize what was going on, and report it?