Looking for Apartment/House to Rent around February 2020

Hello,
My boyfriend and I are considering moving from our current apartment when our lease is up if we can find somewhere comparable in price. Our lease currently ends February 14, 2020. We do have one cat, and are willing to pay a small pet deposit and monthly pet rent if necessary. My boyfriend does smoke, but always smokes outside and is good about discarding the cigarette butts. Our current monthly housing/utilities expenses are around $1,000 a month total for a 1 bed 1 bath, so we are looking to stay around that ballpark. We would like to have someplace to ourselves (no roommates). We may consider a space that is mostly private (own bedroom, bathroom, living, kitchen)--like if someone converted their basement or something into an apartment. Does anyone know of anywhere comparable (not The Gates), or have maybe a duplex of a house they are looking to lease around that time that will accept pets?
Thank you in advance for any leads!
Sincerely,
Sarah