Final answer:
The phrase 'with the oversized hat' in the given sentence is an adjective phrase modifying the noun 'man,' describing a characteristic of the man (his hat). It is not a participial phrase, adverb phrase, or noun phrase in this context.
Step-by-step explanation:
An adjective phrase is a phrase whose head is an adjective. Almost any grammar or syntax textbook or dictionary of linguistics terminology defines the adjective phrase in a similar way, e.g. Kesner Bland, Crystal, Greenbaum. In the sentence "The tall man with the oversized hat had a deep, growling voice," the phrase "with the oversized hat" functions as an adjective phrase modifying "man."
This is because it describes a characteristic of the man, specifically giving more detail about his appearance. Additionally, a participial phrase is a verb that ends in -ing or -ed and acts as an adjective or adverb.
However, in this case, "with the oversized hat" does not contain a participle and strictly functions as an adjective phrase providing more information about the noun "man." Therefore, the correct answer to the question is option B. It is an adjective phrase modifying "man."