She reread the typed note, smiling.
The phrase "the typed note" is serving as a direct object to the verb "reread" as it raises the question of what. While the participle "smiling" is telling the manner how she was reading the typed note, so it is an adverb of manner. Adverbs of manner usually take place right after the verb they serve to. Can we write it like:
She reread smiling the typed note.
The use of the comma before the word 'smiling' also refers to its displacement.