Well, one possibility is that Hermione uses her intelligence to outwit the escaped Death Eater. Maybe she sets up a series of magical traps. She could use her knowledge of spells like the Stunning Spell or the Binding Charm to keep the Death Eater at bay until help arrives.
Well, one possibility is that Hermione uses her quick wits and magic skills to defend herself. She might immediately cast a stunning spell or a protective charm. For example, she could use the Protego spell to create a shield between her and the Death Eater. Then, she would likely try to find a way to escape or call for help. Maybe she'd use her Time - Turner (if she still has it in the fanfiction context) to go back a few minutes and avoid the encounter altogether.
In a fanfiction scenario, perhaps the Death Eater has a change of heart when he sees Hermione hiding. He could be secretly against Voldemort's ways and decides to help her. They might form an unlikely alliance, with the Death Eater protecting Hermione from other dark wizards. Or, the Death Eater could be under a spell or curse that makes him act differently when he encounters Hermione, leading to a very strange and unexpected encounter.
In such a fanfiction, Hermione might initially be in great danger. The Death Eater could try to capture her to use as leverage against Harry Potter. Maybe she uses her quick wits and vast magical knowledge to outsmart the Death Eater at first encounter.
Another idea is that Sirius, as a Death Eater, has some information that Hermione needs to defeat Voldemort. She has to convince him to give it to her while also dealing with his newfound allegiance. It could be a race against time as other Death Eaters suspect Sirius of being unfaithful to their cause, and Hermione has to protect him while also getting the crucial information.
She might react with caution at first. Given that Death Eaters are known for their cruelty, Hermione would be smart to be on her guard. She would likely start by gathering information about the escaped Death Eater, like his known abilities and any possible weaknesses.
In some fanfictions, Hermione might be a Death Eater spy due to a complex backstory. Maybe she was blackmailed. For example, if the Death Eaters threatened to harm her Muggle - born parents, she could be forced to spy for them. This would be a huge deviation from her canon character, but in the realm of fanfiction, it allows for exploring different character arcs and moral dilemmas.
In fanfiction, when Hermione talks to a Death Eater, she might try to persuade the Death Eater to abandon their dark ways. For example, she could use her intelligence and knowledge of the wizarding world to show the Death Eater the error of their beliefs. Maybe she would bring up the harm that Voldemort has caused and how following him is only leading to destruction.
Bellatrix Lestrange is also a likely candidate. Her encounter with Hermione would be extremely intense. Given her loyalty to Voldemort and her rather unhinged nature, she would likely pose a serious threat to Hermione, but Hermione might also find a way to use Bellatrix's madness against her.
Well, in such a fanfiction, perhaps the Death Eater could initially be reluctant to help Hermione. But if there's some common goal, like stopping a greater evil in the wizarding world that might affect both the good and the Death Eaters' side, they could form an uneasy alliance. For example, a powerful dark magic artifact is about to be misused by a new dark wizard and Hermione realizes she needs the inside knowledge of the Death Eaters to stop it.
She could be a target. The Death Eater Inner Circle is known for their dark magic and evil deeds. Hermione, being a smart and powerful witch, might be seen as a threat to their plans. So they could try to capture or eliminate her.