Civitavecchia Overnight rental car parking ?

[medskis]

Hi there,

 

We are renting a car at FCO in mid-March and will do a little touring on the way to Civitavecchia. We’ll have arrived at FCO the prior evening and staying at the Hilton closest to terminals.  No need to tour Rome again.  
      While we can board our Windstar ship same day we pick up the car, the ship remains in port until the evening of the following day.  We’d like to leave/park the rental car on day 1 and use it day 2 to go north to tour some of the inland hill towns /have lunch as we did in 2018 (on the way to Siena).  Looks like Hertz or Budget coming up as primary choices (in 2018 we had Europcar).  Just wondering if anyone has parked a car overnight near the port/or at the local rental company property and continued their rental the following day.  
 

While I like Hertz as a rental source, I believe their property is not as walkable to the port as Budget/Europcar are. I’d like to get the rental all set soon.  Thank you!