Patnem is a gorgeous beach to the south of Palolem and Agonda. It is quieter than Palolem, with fewer travellers passing through, and many longer-term residents. With a smaller bay than both Agonda and Palolem it feels more cozy, with enough to keep you entertained without being overloaded with bars and restaurants.
Palolem was one of my most enduring memories of India when I first visited. Practically deserted, at the end of the season several years before it had been 'discovered', it really did feel and look like paradise - a beautiful, sweeping bay fringed by palm trees, with just a small cluster of huts offering basic acommodation and meals. We stayed with a family in one room of their two room house, with nothing but palm trees between us and the beach.