You need to understand that real estate websites (like Trulia, Zillow) are limited with search filters. Realtors who own the MLS have far more search filters than public websites do. Realtors intentionally refuse to list whether rentals accept pets or not because they want you to call them. If they list everything the MLS has, then it means consumers won't call them and they will lose business. Therefore, realtors leave important information off their listings. Examples: whether pets are allowed or not, HOA fees, co-op maintenance fees, etc.
You need to contact some realtors and have them email you listings that accept small pets.