I have checked with Cicar, one of the cheapest and reliable companies in Canary Island, and they told me that you can take the car to other island as long as you return it in the same island where you rented it. You don't have to pay extra money for this but you have to ask them for an authorization.
This is what they answered to me about you question. This is the answer they gave me. It is in Spanish but basically It says what I told you.
From: <[email protected]> (sent by [email protected])
Usted puede alquilar un coche en una isla y pasarlo a otra, siempre y
cuando devuelva el coche en la misma isla que lo recogió, sin cargo
extra. Además deberá pedir la autorización a nuestra oficina para que
conste en su contrato y tenga una cobertura total.
On the other hand, Cicar is an inexpensive company and they have offices in every dock so you can take a new car as soon as you arrive to the next destination. This option can be the cheapest since you don't have to pay for the car transportation when you move between islands, which is not usually cheap.