Symmetric Searchable Encryption (SSE) is an encryption technique that allows users to search directly on their outsourced encrypted data while preserving the privacy of both the files and the queries. Unfortunately, the majority of the SSE schemes allows users to either decrypt the whole ciphertext or nothing at all. In this paper, we propose a novel construction based on traditional symmetric primitives, that allows data owners to bind parts of their ciphertexts with specific policies. Inspired by the concept of Attribute-Based Encryption (ABE) in the public setting, we design a scheme through which users can recover only certain parts of an encrypted document if and only if they retain a set of attributes that satisfy a policy. Our construction satisfies the important notion of forward privacy while at the same time supports the multi-client model by leveraging the functionality of SGX for the synchronization of users. To prove the correctness of our approach, we provide a detailed simulation-based security analysis. Finally, this analysis is coupled with a rigorous and extensive experimental evaluation that shows the effectiveness of our scheme.