En refers to the whole expression autre bonnet (que celui-ci), not just que celui-ci.

Without en, the phrase becomes:

elle ne voulut plus porter d’autre.

This indeed is incomplete because there is no indication about what autre is referring to. Maybe does "She no longer wanted to wear some other" work in English but in French, we need to tell some other "what".

Without d’autre but with en, the sentence is grammatical:

elle ne voulut plus en porter.

but its meaning is different. She no longer wants to wear any bonnet at all.

Upon re-reading a part of the book "The Structure of Modern Standard French" by Maj-Britt Mosegaard Hansen, I realize that I grossly misunderstood the use of en. It is far too simplistic to say that en usually replaces "de + _____".

The following excerpt applies to my question (but there are other uses of en discussed in the book) :

In addition, en is frequently used to represent only the head noun
of an indefinite noun phrase, leaving both the determiner and any
modifiers behind, as shown in (30)-(31) below. This is possible no
matter what kind of indefinite determiner is used:

(30) Un train peut en cacher un autre. ( = Un train peut
cacher [un autre train.]) (31) A: Je pourrais te piquer une
cigarette, s’il te plaît ? B: Oui, mais je n’en ai que des
légères. ( = Oui, mais je n’ai que [des cigarettes légères.]).
(from Section 17.2.2 in "The Structure of Modern Standard French")

It is worth reminding myself that some indefinite determiners are:

• Indefinite Article
• Partitive Article
• Numerals

(from Section 12.1, "Definite and Indefinite Determiners")

Given the above information, how en is used in the sentences below now makes sense. The only thing I still must do, is to make sure I understand why "d’autre bonnet que celui-ci" and "des jolies perles" are considered indefinite noun phrases.

• Elle ne voulut plus porter d’autre [bonnet que celui-ci]
  Elle ne voulut plus en porter d’autre

• Elle portait des [jolies perles] roses.
  Elle en portait des roses.