She only went at the suggestion of the Muggle because she thought it would seem strange if she followed up later to discover she never arrived. Maybe she had a dear friend who worked in the shelter and she would casually ring her up to check to see if Hermione ever made it. There might have been laws and societal expectations put in place to protect the identity of the poor souls seeking refuge, but Hermione knew enough about human nature to understand that even the most proficient and law-abiding person could sometimes bend the rules. If her unexpected savior suspected that her uninvited guest was on the run from something other than an abusive husband, Hermione might have found herself the person of interest in another search.