It is challenging to convert natural language (NL) questions into executable structured query language (SQL) queries for text-to-SQL tasks due to the vast number of database schemas with redundancy, which interferes with semantic learning, and the domain shift between NL and SQL. Existing works for schema linking focus on the table level and perform it once, ignoring the multi-granularity semantics and chainable cyclicity of schemas. In this paper, we propose a progressive schema linking with multi-granularity semantics (PSM-SQL) framework to reduce the redundant database schemas for text-to-SQL. Using the multi-granularity schema linking (MSL) module, PSM-SQL learns the schema semantics at the column, table, and database levels. More specifically, a triplet loss is used at the column level to learn embeddings, while fine-tuning LLMs is employed at the database level for schema reasoning. MSL employs classifier and similarity scores to model schema interactions for schema linking at the table level. In particular, PSM-SQL adopts a chain loop strategy to reduce the task difficulty of schema linking by continuously reducing the number of redundant schemas. Experiments conducted on text-to-SQL datasets show that the proposed PSM-SQL is 1-3 percentage points higher than the existing methods.