Try Levanto: a nice town that is part of the Cinque Terre park region (one town north of Monterosso), covered by the trains pass and hiking pass (if you want to hike between Levanto and Monterosso - I've never done it, sounds like a rewarding but semi-difficult hike). Park the car in Levanto and take the train to the villages.
I've stayed in Levanto twice. It's a nice beach town, even if not quite as the dramatic "built into the side of the hill" feeling of the five main villages.
Use Booking.com to find places - use their map feature. Try to find a place within walking distance of the train station. The trains are frequent and easy between villages. (There are boats, too, but assume the trains have much more frequent schedules.)
If you can't find anything in Levanto, try going further north. Sestri Levante is a nice little town (once you get away from the train station area). Not as nice as Levanto and further north by train (and not part of the park region) - but you can park here too and leave the car in town.