I would never assume anything with the present state of DL's IT.
FWIW, the Domestic CofC does not show an exemption for "or next available flight"
RULE 180: STOPOVERS
A) Stopovers will be permitted only upon payment of the combination of applicable fares, or stopover charges.
B) A stopover, as used herein, will occur when a passenger arrives at an intermediate or junction transfer point on a flight on any carrier, and fails to depart from such intermediate or junction transfer point within 4 hours after their arrival at such point.