Yes. That is one of the features that makes the Chairman such a beast to fight.
As long as it is during the phase that "activates" that card (in this case, an Underboss' "End of turn" effect that puts into play a particular kind of Thug with an "End of turn" effect), then the new card also takes effect, since, in absence of player prerogative, cards activate in the order of play. The newest card goes into play last, so it activates last.
A number of the villains utilize on this "play a card during the end-of-turn phase that activates in the end-of-turn phase" chain to deal a hefty amount of hurt. This is one of the reasons taking out Villain cards that play more Villain cards in the end-of-turn should be a priority.
As long as it is during the phase that "activates" that card (in this case, an Underboss' "End of turn" effect that puts into play a particular kind of Thug with an "End of turn" effect), then the new card also takes effect, since, in absence of player prerogative, cards activate in the order of play. The newest card goes into play last, so it activates last.
A number of the villains utilize on this "play a card during the end-of-turn phase that activates in the end-of-turn phase" chain to deal a hefty amount of hurt. This is one of the reasons taking out Villain cards that play more Villain cards in the end-of-turn should be a priority.