I believe the term "subject complement" has gone out of favor. These days, ill would be referred to as a predicate adjective, for more about which see Predicate Adjective explanation in layman's terms.
Regardless of whether "subject complement" or "predicate adjective" is the term you use, however, neither ever modifies the verb. The predicate is the subject of the sentence, and that's perhaps why the term "predicate adjective" has come to be used. Ill is an adjective describing Liam, the subject (predicate) of the sentence. Very is an adverb describing ill.
Incidentally, I believe you can have a sentence which reads simply "Liam is." For example:
John: Who is it that is very ill?
Jane: Liam is.