Agent skipped all of Riddler's dialogue, so when Linda stormed out in a rage, he had no idea what had happened. He wanted to ask, but figured he would only get more riddles in return, so he decided to take action instead.
He followed behind Linda without being noticed, of course. This kind of stalking skill might not have fallen within his Character Design, but the boundaries of such things were vague. After all, everyone could follow someone; it was merely a question of whether they got caught. It was not as though it required any special technique.
In fact, Linda was so agitated right now that even without using any technique, Agent would have been hard for her to notice. So he did not lose any health.
Before long, however, he discovered that Linda was heading for the cemetery. That gave him pause—not because he was worried about being discovered, but because going to a cemetery when there was definitely an evil spirit around was truly not a good choice.
After thinking it over, Agent still followed her. It was daytime, after all, and he had gone to the cemetery during the day before without triggering any sort of hunt. He had gone with Linda then, too, which basically eliminated all the variables. So he followed her there.
Only after arriving did he discover that Linda had somehow gotten hold of a shovel and was digging up a grave.
Watching from afar, Agent realized that Linda seemed to be digging up her father's grave. She was incredibly strong, and in just a few strokes, she had dug the grave open. Then she took something out, put it into a bag, and headed back.
After she left, Agent did not dare linger in the cemetery alone, so he followed her