She wanted out || of this dump||. That is my parse.
To want out is a phrasal verb meaning: to not want to be in a place or relationship. It can be followed by a prepositional phrase or not.
The opposite also exists: to want in, meaning: to want to be a part of something.
They were playing poker and I wanted in [to the game].
To want out + of this dump is a phrasal verb + a prepositional phrase.'
In English, you are in or out of a place or relationship. And you can want in or into a place or relationship.
The cat wanted out of the house. = The cat did not want to be in the house. or: The cat wanted to go out of the house.
For me: to want in/into or want out (to desire to come, go or be or the negative of that: Merriam Webster, intransitive) can be followed by prepositional phrases that act as a complement to the idiomatic usage of the verbs.
Intransitive verbs can be followed by a prepositional phrase or an adverb to add to the thought being expressed, but they can never be followed by a noun, which would act as the object of the sentence.
"of this dump" is a prepositional phrase that adds to the thought being expressed. Here, expressing the place or location from which relief is sought.