Can you hang some flypaper strips outside the cage? We had a moth problem in our mudroom last summer and a few days of hanging the flypaper got rid of them FAST.

The flypaper strips are CHEAP - like $2 for 4 strips. They are the kind with a sticky glue that is rolled up inside a small tube. Completely pet safe, but if you stuck it in the cage, you'd have a sticky chin.