The word "weiter" does not require "sein".
"unterwegs sein" on the other hand is a composed verb. See Duden https://www.duden.de/rechtschreibung/unterwegs_sein. You have to use the verb "sein" no matter if you use it in combination with "weiter" or not.
E.g: "Ich bin noch immer unterwegs", "Sie ist unterwegs"
But "unterwegs" by itself is just an adverb indicating where you are (= lokales Adverb).
So you could call your boss asking where she is, and she could answer:
"Ich bin unterwegs"
This means roughly "I am on the way".
It is similar to "Ich bin am Weg" - it describes where you are. Just as you could say: "Ich bin schon in der Arbeit."
But you could ask your friend where she bought her sandwich and she could answer: "Ich habe das Sandwich unterwegs gekauft." In this case you don't need the verb "sein" because "unterwegs" only indicates where the sandwich was bought. The action of the subject is described by the buying of the sandwich while the "unterwegs" tells you where the action took place.
In the sentence "Ich bin weiter unterwegs." the "unterwegs sein" describes the action of the subject.