Here is a list with links to car hire companies that have an office in the Reina Sofia airport in Los Cristianos, most commonly known as Tenerife South:
Orcar Canarias (from our own experience this is actually the cheapest)
Autoreisen (They actually share an office with Orcar in TFS Airport)
Avis Great service in general, although they are one of the most expensive car hire companies here.
Cicar They only operate in the Canary Islands.
Europcar They usually have the cheapest prices in his category (worldwide rent a car company)
Hertz Great service, although they are one of the most expensive ones
Thanks for the useful info. It’s so interesting