I recently faced the same problem and decided to find a reliable car rental service. I came across
https://findycar.com/car-rental-madeira/ — specifically for Car Rental Madeira. There, you can easily find a car for every taste and budget, choose additional options, insurance, and pickup location, as well as compare prices from different rental companies. We chose an economy class car for our family, picked it up at the airport, and it literally changed the whole trip: now we plan our own routes, can visit mountain villages, beaches, and viewpoints, and are not tied to bus schedules. The cost was reasonable, and the booking process was very simple — everything was done online, with confirmation by email. I recommend checking out Car Rental Madeira on the website; it really simplifies travel and gives you freedom of movement.