Tenant moved out before lease term

[Ping W.]

The current lease of the tenant expires the end of October but they moved out at the beginning of October. They did pay October rent in full. We are looking for new tenant now. If I find a tenant before the end of October, do I need to refund the previous tenant's partial rent for Oct? They did give me 30 day advance move out notice.

A property management agent told me the tenant abandoned the house when they moved out. Is that true? What is the condition to say tenant abandoned the house.

I am new as landlord. I appreciate any input from all your expert!