Just any old appropriate adverb won't do; there should be some wordplay, too.

Agreed. Words beginning with in- or un-would seem to be likely hunting-grounds, as in

"I just gagged on my drink," said Tom insipidly.
"I'm telephoning you from a Greek island," said Tom indiscreetly.
"Is this how you serve as an artist's model?" asked Tom indisposedly.

"The horse has gotten loose," said Tom unstably.
"I'm out in my boat today," said Tom unassailably.