Fenced backyard. Does that mean you have pets? Many rental properties will not allow pets so that may reduce your choices even more. You need to disclose pets up front when inquiring about a rental home if you do have pets.
It will be difficult to find something in the next couple of weeks because most owners will not hold their property off the market for 30+ days to wait for you to move in. You will probably have better luck starting to search, in earnest, about 30 days before you need to move in. Always good to check earlier though in case you find an tenant occupied listing that won't be ready for a new tenant for 30 to 60 days. These are rare.
Good luck with your search!