Why is "zur Schule" inside the verbal bracket here?

Question

Manchmal, wenn er mit diesem lächerlichen Haarschnitt partout nicht hatte zur Schule gehen wollen, hatte er es geschafft, dass sein Haar rasch nachwuchs…

I'm trying to learn German and I recently picked up reading; as a result I get exposed to a lot of new vocabulary.

I'm used that at the end of a clause (excluding Nachfeld) there're only verbs. Could it be that zur Schule forms a pair with gehen, the kind that acts like a separable verb, and that's why, to my surprise, there is something else than verbs at the end of this clause?

Answer 1

Your sentence shows the Sonderregel of the ordering of infinitive verbs.
When the second infinitive verb is reigned by a modal verb, in your case wollen, the auxiliary haben is placed directly to the left of the two:
(Duden Grammar Nr.684)

, was sie nicht hatte machen wollen.

Duden Grammar Nr 686 tells us that if the Sonderregel is used, a constituent that is closely connected to the infinitive main verb can be pushed between haben and infinitive main verb:

, was sie nicht hatte in Anspruch nehmen müssen.
, was sie nicht in Anspruch hatte nehmen müssen.

, weshalb sie nicht hatte zur Schule gehen wollen.
, weshalb sie nicht zur Schule hatte gehen wollen.

It does not change any meaning.

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Wenn is used as a temporal subjunction, like English "when".

Answer 2

I have also found the answer myself now encouraged by your answer in an English source:

Hammer's German Grammar and Usage 7th edition
19.1.3b(v) and 19.7.2 of which does also mention that "these phrases" are part of the verbal bracket (besides what you mentioned).

Feels bad when I could've found the answer myself in a book I own but for some reason I didn't imagine I could find the answer at the section of "verb order in the final position".