Clint stepped out just in time to see Peggy run. He'd talked to Tony, heard what happened on the CB, knew enough to realize who it was but not enough to know if Maria was really safe or not. He couldn't imagine Peggy, even if she was the Winter Soldier, hurting a baby without orders to do so but he didn't quite know what her plan actually was, and that terrified him.
He had just enough sense to grab a gown, putting it on as he moved as quickly as he could after her. He knew right away that it would never be fast enough to catch her, he couldn't move nearly as fast as he usually could and she'd probably be able to beat him even then. But he knew what he could do best and how to use that to his advantage. The hospital was small enough that if she hid here they'd be able to find her, but if she got away from it they'd have hell searching the island if they didn't even know where to start.
He went straight for the elevator, hating that he couldn't dash up the stairs but he knew it'd take much longer now. If he could just get to the roof before she got out, he'd be able to see where she went and send someone in better shape after her.