If you really wanted to add a day or two, you could drive from Verona to Lake Garda which is out of the way but not terribly so and would have some pretty roads to drive on plus hotels will have parking.
Otherwise, your plan of a car doesn't make much sense if the only purpose is to get you from Verona to La Spezia to drop off a the car and catch a train there, when you could just catch a train from Verona and avoid the car.
Don't think there are any pretty Italian countryside roads on your way if that is what you had in mind, looks like mostly highway which is not really something memorable vs. taking the train to save time, money, convenience.
Any of the cities you might want to visit along the route or a little off route like Padua, Parma, Bologna, Modena will have ZTL restrictions and having a car for that one night stay will be a hassle there as well.
Keep in mind where you would be renting/dropping off a car is rarely right where a train is so you have logistics of dropping off the car and then getting to the train station which also adds time and money.
Not to mention, there is some fear visiting an area in Italy while your car is parked somewhere with your luggage.
The trains from La Spezia to CT run really late, so no really time worries.
Just wait on the car until you are leaving Florence and use it to visit Tuscany and drive through Umbria ;stretches of roads there will be very scenic and I assume you would take a few days to get down to Sorrento.
It should not be a problem to get a 1 day rental pickup in Florence and drop off in Sorrento a few days or week later
