A searchable symmetric encryption (SSE) scheme is a method that searches encrypted data without decrypting it. In this paper, we address the substring search problem such that for a set D of documents and a pattern p, we find all occurrences of p in D. Here, a document and a pattern are defined as a string. A directed acyclic word graph (DAWG), which is a deterministic finite automaton, is known for solving a substring search problem on a plaintext. We improve a DAWG so that all transitions of a DAWG have distinct symbols. Besides, we present a space-efficient and secure substring SSE scheme using an improved DAWG. The proposed substring SSE scheme consists of an index with a simple structure, and the size is O(n) for the total size n of documents.