What new insight could be produced by stylistics or linguistic analysis apart from the normal 'fragmentation, stream of consciousness' in analyzing Mrs Dalloway?
Indeed it is true that the stream of consciousness is the most salient element in Mrs. Dalloway, which, as a sample of modernist literature tends to gear toward the introspect world of the character, moving the plot from the inside out, rather than from the dialogue and then into the mind of the character. Like your question also points out, it is true that the stream of consciousness is characterized by the fragmentary way that the narrative flows: the present, the past, and the future do not follow each other sequentially.
Other insights aside from those above are suggested by Dowling (1991) (who quoted Robert Humphrey) in Mrs. Dalloway, Mapping Streams of Consciousness. From Dowling's analysis we can draw out the following techniques that could serve as foundations for analysis leading to new insight based on the stylistics of the novel.
direct and indirect interior monologue- Clarissa's direct monologue is elicited by thoughts (e.g., purchasing the flowers herself at the beginning go the story: "It is probably the Queen, thought Mrs. Dalloway"). However, there is interior monologue in the thoughts that surface seemingly out of nowhere but still relate to the different characters' states of mind (e.g., Clarissa's musings in the shop: "she began to go with Miss Pym from jar to jar, choosing, nonsense, nonsense, she said to herself, more and more gently").
Another example of direct monologue would be:
She’s a queer-looking girl, he thought, suddenly remembering Elizabeth as she came into the room and stood by mother. Grown big; quite grown-up, not exactly pretty; handsome rather; and she can’t be more than eighteen. Probably she doesn’t get on with Clarissa.
However in the same passage from which that is quoted we find evidence of indirect monologue that is made to look like direct speech. The reason for this is that the indirect monologue recalls a memory of something that may have actually occurred or that the character may have wished to have occurred, like in Peter's thoughts below for nowhere in the text does Clarissa say "There's my Elizabeth":
‘There’s my Elizabeth’ – that sort of thing – why not ‘Here’s Elizabeth’ simply? – trying to make out, like most mothers, that things are what they’re not.
blurred consciousness - Notice how Clarissa's memories shift with those of Peter Walsh's to form a composite, a unity. The blending of their thoughts is a form of consciousness blurring, a technique within stream of consciousness, as in the episode about the dog: "It was as if she said to Peter--it was all aimed at him, he knew...."
effacement of personality - As a society woman, Clarissa has to show different faces to different people. She is expected to fulfill the societal roles that women of her class, and her time, are meant to do. Hence, her personality acquires different modes; one for Peter, one for herself, one for the rest of the world.
We could also include ephemerality in the form of evanescence, and the brevity of each stamp that Clarissa and the other characters provide with their own interior monologues which lead to soliloquy and more internal thought.