Car rental agency from Madrid airport

We’ve used both Enterprise and Avis on previous trips. We’ve had good experiences with both. A smaller, or Spanish, company may not have an office where you want to drop off the car, so you may have to go with a larger company.

Sorry to be that guy, but are you planning to drive from Madrid to Granada after a flight from another continent? It is a fairly long drive to undertake in such circumstances: I would reconsider.
If you are coming from closer, no problem!

If you're spending a few days wandering your way down, that's one thing. But if you're looking to drive directly, to then pay for parking in Granada, a train would probably be cheaper, easier, and to Balso's point, perhaps wiser.

Personally I like to use Sixt for car rentals in Europe, including Spain, because you can purchase “zero excess” (zero deductible) LDW coverage. With other rental companies, the LDW coverage still includes a sizable deductible, so if you get any small damage on the car, including damage that may not be your fault, the rental company can charge you many hundreds of euros and you have no recourse.

What roger said isn’t exactly correct. Liability insurance is mandatory for European rentals and is included in the rental price. That’s why the rental agencies don’t go over a returned car with a fine tooth comb like they used to. There are some exclusions, but minor scratches and dings are nothing to be concerned with. A few years ago somebody backed into my rental in Sevilla leaving a grapefruit sized dent in the fender. We dropped the vehicle off and never heard a word. Where the excess coverage comes in handy is when damage to the vehicle is because of something you did, e.g., scrape the vehicle’s side driving down a really narrow street, backing up into a bollard and bending the bumper, etc.

I’ll have to disagree somewhat with jaimeelsabio. Yes, you’re automatically covered for liability, and the Loss Damage Waiver (LDW) limits your costs for damage caused by you. But also if your rental gets scratched or hit when you’re parked, you’ll be responsible for that damage. I’ve read plenty of complaints from people about getting exorbitant charges for minor damage - maybe scams. If you don’t get charged, you’re lucky. I prefer to get LDW with zero excess for peace of mind.