Antagonist vs someone else
Question: This may be a weird question; but is there any chance that an antagonist, though is against the protagonist and overall story goal, to have more of a personal vendetta against a secondary, supporting role on the protagonist side?
As long as the antagonist is against the story goal being achieved, there's no problem.
For example, you could have a story where the antagonist has a vendetta against an innocent person. The protagonist could have the goal of protecting this person. It seems to me there are several Westerns starring John Wayne that work this way, with him protecting a young girl. Similarly, in various versions of The Three Musketeers
, the hero has the goal of protecting the French Queen/King against her enemies who are trying to depose the monarchy.
You could also have a story in which the antagonist's dislike of the protagonist's ally (spouse, friend, mentor) makes him automatically predisposed to thwart the protagonist. In other words, the dislike could be a source of motivation for the villain.