Car rental on Naxos

You may want to check with your hotel for this info. We stayed in Naxos before Covid and there was no need for a car. The hotel picked us up from the airport which is quite close to town and we were able to walk everywhere within the town, it is quite small. To see the rest of the island we took a tour with a local guide who provided the transportation.

Renting a car on an island is the best way to visit it without being dependent on buses or taxis (and go where the buses don't go)., especially on Naxos which is the largest island in the Cyclades, we stayed there almost 2 weeks (over several stays) renting a car and we've only seen part of it.
The road network is well maintained, without any particular difficulty. Most of the time you don't have to worry about dirt roads if you use them, even buses pass there. An example here of the road that runs along the west coast:

https://maps.app.goo.gl/hjpqqdFV49x4P72MA

Preferably follow the road signs rather than your GPS which could take you into unnecessary shortcuts

But if you plan to only stay in "town" in Naxos Town and just go for a day trip to one place that's another approach, it's up to you.

There are quite a few local car rental companies in Naxos. It is always best to book in advance, there is no point in booking months in advance. 3 or 4 weeks is more than enough in October. If you are 6 people it might be possible to rent a minivan instead of 2 cars, I don't know.

Some local companies:

NaxosRentACar.com

Naxos Ride Rent A Car

Fun Car and Rides

New Car Naxos - Rent a Car..

etc., etc

Do a "rental car" search on Google Maps, you will find all the details and websites

At least when we were in Naxos in 2018, there was no option to rent a mini van. We had a group of 7 and rented two cars. We did so through the hotel we were staying at.

All above advice is good. But I will tell you how we did it.

We stayed the first two nights on Naxos in the old town which was walking distance from the ferry. From there it was a short walk down a main street to St. George's Beach. While walking to that beach we saw numerous rental businesses. I just popped into one and said, "Can I arrange for a car tomorrow morning for four days?" And they said, "Yes". So I said, "See you tomorrow at 10?" She said, "Yes."

I handed over my passport, driver's license and credit card and was out of there in less than five minutes. When I ambled by at 10 the car was ready and had been maneuvered in such a way as to have been seemingly impossible but which allowed me to just drive straight out of there.

Naxos is lovely by car, just grand.

One thing we have noticed in Greece is the rental agencies stay open quite late at Night. Next morning after a short night sleep they are not quite as efficient as one would expect and it can be a slow process getting the car next morning. Even with a pock up time at 9 we often did not get the car until 10. Since the car is rented on the basis of 24 hours we started to rent the car in the evening and had it available as early as we wanted next morning.
Here are some images of our driving tour of Naxos.
Trip around Naxos https://www.flickr.com/photos/stanbr54/albums/72157634605629689