I've stayed in the Cinque Terre three times, and I've actually stayed in Levanto each time. It's a nice town - not the "postcard charm" of the other five towns but still a pleasant town. It's very convenient by train to destinations up and down the riviera including the CT towns. The CT train pass includes Levanto.
On my last visit last May, I hiked one day from Levanto back to Monterosso (does not require the CT hiking pass, for what it's worth). This hike was not very crowded at all. It's not the best hike in the Cinque Terre, but it has some great moments. At one point, just above Monterosso, there are stunning views south down on all five CT towns.
On another day, I did take the ferry from Levanto, but I took it all the way past the five villages to Porto Venere, a nice town not connected to the others by train. It had plenty of tourists but because it's not part of one of the major hiking paths, it was probably less crowded than the others.
So on your off day you could certainly do one or both of the things I did and on the other do some of the other hikes. You can hike between Monterosso and Vernazza and between Vernazza and Corniglia (assuming those two trails are still open) - these are the most popular hikes. You could also do a much more difficult hike (but breathtakingly scenic) between Manarola and Corniglia - with a steep detour up to the village of Volastra, with a zillion steps up, up, up...and then up some more! I did this one in May too - it was more crowded than the hike from Levanto to Monterosso but surely much less crowded than the main hikes. This may have been the most rewarding hike (regarding views, anyway) of all the CT hikes I have done.
The suggestion to go north to Santa Margherita Ligure etc. on one day is not a bad suggestion either - some beautiful towns up there. Up to you what you want to do.