Renting with a small dog in an apartment in inner west Sydney

I am wanting some advice on renting in the inner western suburbs of Sydney with a small 5 kg dog. I have read that most pet-friendly rentals are around the Mascot area, which would be perfect because I have a job offer at Sydney airport. When I select ‘pet-friendly’ on the realestate.com search function for Mascot and surrounding suburbs it brings up about 70 different properties, which seems quite a few. However, I am concerned about how many other people would also be applying for pet-friendly rentals and what the competition for the property is likely to be like. I would not want to find myself in a position where I was desperate to find somewhere to live and had to re-home my dog, and if this is a possible outcome then I will decline the job offer.