A travel agent suggested La Spezia for two nights...
As an aside: I would regard any travel agent who suggested that with a great deal of skepticism. This strategy - going to the trouble and expense of sleeping just outside the CT towns, and doing day-trips into the towns themselves - suggests that this travel agent does not seem to understand really basic facts about today's tourist experience.
The CT towns are mobbed and overwhelmed by day-trippers every day, during the core daytime hours. The way to mitigate that is to spend the night IN the CT towns, not outside them: because before and after the day-trip crowds fill the streets, the towns retain some of their pre-overtourism charm. When crowds of tourists are clogging every square inch of the towns, it's just a battle of waving selfie sticks and jostling for the "picture spots" with not a lot of the original charm.
The basic strategy for dealing with places that have become too popular is to be there when the crowds thin out, and not to go there along with the rest of humanity, as mid-day day-trippers. Your travel agent is giving you advice that is the exact opposite of what most savvy travelers want. Staying in La Spezia means you still have to pay for a room nearby (it's not going to be a whole lot cheaper than staying in one of the towns itself), and as a day-tripper yourself, you then suffer the same overtouristy crowds. It's the worst possible combination.
Stay in one of the CT towns, get up early (very early) and enjoy the places before the cruise ships and trains deposit thousands of day-trippers. During mid-day, when the crowds are most intense, get out and go for a hike. Come back late afternoon, as the day-trippers drain away. Then enjoy a few hours in the evening when the worst of the crowds are gone. The opposite of what you would get following the advice of your "travel agent." Just one person's opinion.