In my opinion it is not absolutely false, but it sounds "bumpy".
The phrase "Meine Mutter sah betrübt und besorgt aus, und sogar hatte meine Tante einen Ausdruck" consists of two main clauses (Hauptsätze). The first part "Meine Mutter sah betrübt und besorgt aus" is correct, but the problem is the second part
(und) sogar hatte meine Tante einen Ausdruck
Although German word order is variable, everybody would expect
(und) sogar meine Tante hatte einen Ausdruck
in SPO structure. However, the problem is related to the word sogar. If we replace it by the adverb heute, we get
Meine Mutter sah (heute) betrübt und besorgt aus, und heute hatte meine Tante einen Ausdruck
and I can't see anything wrong with it. In fact,
Meine Mutter sah (heute) betrübt und besorgt aus, und heute meine Tante hatte einen Ausdruck
is definitely false.
An alternative (and indisputably correct) formulation of your sentence is
Meine Mutter sah betrübt und besorgt aus, und es hatte sogar meine Tante einen Ausdruck
Update:
"Sogar" is a grammatical particle. Quotation from English Wikipedia:
In grammar, the term particle (abbreviated PTCL) has a traditional meaning, as a part of speech that cannot be inflected, and a modern meaning, as a function word associated with another word or phrase to impart meaning. Although a particle may have an intrinsic meaning, and indeed may fit into other grammatical categories, the fundamental idea of the particle is to add context to the sentence, expressing a mood or indicating a specific action.
[...]
Particle is a somewhat nebulous term for a variety of small words that do not conveniently fit into other classes of words.
German Wikipedia says
Man rechnet zu den Partikeln – im weiteren Sinne – alle nicht flektierbaren Wörter einer Sprache (wie Adverbien, Interjektionen, Konjunktionen, Präpositionen) oder – im engeren Sinne – nur solche nicht flektierbaren Wörter, die nicht den Unterklassen Präposition, Adverb oder Konjunktion angehören.
That is, the class of particles includes adverbs, interjections, conjunctions and prepositions. It seems that the status of "sogar" is not absolutely clear-cut. Its function is to emphasize something which comes a little unexpected and may be attributed to verbs, nouns and adjectives. Examples:
Er kann sogar tanzen. [sogar + verb]
Er kann sogar hervorragend tanzen. [sogar + adjective]
Sogar er kann tanzen. [sogar + noun]
If it refers to a noun (as in sogar meine Tante), then the word order is determined exclusively by the noun. You can make a test by omitting "sogar" and check whether the remainder is correct. The sentence hatte meine Tante einen Ausdruck is not correct.