They were just really brutal with their response, which was completely out of character for SHIELD from previous films. Didn't really help HYDRA at all and it just motivated Captain America to find out more about the deeper conspiracy at play.
They [Hydra] were pretty smart for the last few decades, working in secret....then they just messed it up at that point.
Hydra didn't know how much Cap knew and couldn't take a chance. And since they made it look like Fury had gone rogue, Cap's close ties to him helped them to justify the man hunt.
Edit: what I did wonder was the dissonance with the speech by Pierce about how people are scared enough to be willing to give up their freedoms, when they failed before by trying to force it. And then their master plan is to kill millions of people, including a whole bunch of US citizens, on US soil, so that they wouldn't be a threat. Ok they could try to spin it by shouting "oh shit, domestic terrorism!", but come on... In fact they would have needed something like Cap attacking the helicarriers for it to even make a lick of sense.