I couldn’t explain the grammar behind this and am answering purely by intuition (I am a native french speaker).
“En avez-vous d’autres” means “Do you have more of them”. Leaving out “En” and saying “Avez-vous d’autres” wouldn’t work. The best comparison I can come up with is saying “Have you more?” in english.

“En” is referring to the prisoners. The governor has already been visiting several prisoners and he is asking to the director if there are more of them to visit:

Do you have more prisoners (that you plan me to visit)?

Edit: Here is the original Le Comte de Monte-Cristo chapter.

En can be several things in French (preposition, adverb, adverbial pronoun, among others). In the sentence referred to by @jlliagre:

L’inspecteur se tourna en souriant, et dit au gouverneur :
« Je ne sais pas pourquoi on nous fait faire ces tournées inutiles. Qui voit un prisonnier en voit cent ; qui entend un prisonnier en entend mille ; c’est toujours la même chose : mal nourris et innocents. En avez-vous d’autres ?
– Oui, nous avons les prisonniers dangereux ou fous, que nous gardons au cachot.

we know it is an adverbial pronoun because of what comes before :

Qui voit un prisonnier en voit cent ; qui entend un prisonnier en entend mille

and what comes after :

Oui, nous avons les prisonniers dangereux ou fous, que nous gardons au cachot.

If the quote was stopped just after qui entend un prisonnier en entend mille we could understand en as meaning “such/similar prisoners”. But what comes after makes it clear that en does not stand for “similar prisoners” but for “different type(s) of prisoners”, i.e. prisoners who do not only have to say they are innocent or not fed properly. I’d translate the sentence with :
“Do you have a different type of prisoners?”

Doesn’t “Avez-vous d’autres” mean “Do you have something else?”

Avez-vous d’autres is not possible. Something is missing :

Avez-vous quelque chose d’autre ?

Avez-vous des prisonniers différents ?

And if context makes it clear we are talking about “different prisoners” we can use en :

 En avez-vous d’autres ?

This answer has been given after OP posted a question on meta and from context given in @Jiliagre’s answer. The modified text in the graded reader referred to by OP might be different, the exact quote from the graded reader is missing from OP’s question.