Have a rental application for them to fill out. That will eliminate about 1/3 of inquiries because they don't return them. You can also ask for an application fee for a background check.
If the first question they ask is, "What's the rent?" I act like I never heard it. I ask are you calling for yourself or someone else? If it's for them I ask how big the family is, especially right after they as how many bedrooms. I basically never answer their questions in the beginning of this conversation. I find out if they are married, or have additional companions of any orientation, married, divorced, number of kids, where they work, how long they been there. Where are they living now, how long been there, why relocating?
On the application there is space for 3 referrals. I tell them the writing has to be clear so I can read it. The people whose names they gave me need to know I will be calling and if they don't answer the phone after couple of tries I'm not going to keep wasting time, and the numbers need to be current. I always contact their employer and confirm how long they been there, if they show up on time and are considered good workers. After a couple of minutes with the employer I end the conversation by asking if he had a rental property would he consider renting to this person/family.
I don't do it, but I've seen folks state the credit score required to rent, and if pets are part of the family the credit score is like 100 pt. higher!
Believe it or not you get better renters with the maximum rent for that property in your area.
I would not add the 3rd BR. Just adding expense and inviting more wear and tear.