During his sole year with the Oilers, Naponic was third on the depth chart behind Charley Johnson and Jerry Rhome.
Up until the final game for the 1970 season, he had thrown only one pass, for an incompletion in a 44–0 loss to the St. Louis Cardinals.
His final game took place at the Cotton Bowl against the Dallas Cowboys.
With injuries to Rhome and Johnson, Naponic took over the quarterback duties against the Doomsday Defense.
[2] Naponic's son, Jackson, later played football as a fullback at Texas Christian University while pursuing a finance degree.