What is the price range? There are houses in the Bronx that are currently on the market that would include a small yard. Prices for single family homes, that match your criteria, range from $289,000 to $745,000. You might also want to consider Yonkers or Mount Vernon. It is a short drive down the Bronx River Parkway. Single family homes start at $279,900. There is also New Rochelle. Prices there for single family homes start at $335,000 and you are the 15-20 minute driving range of your job.
Here are some articles from the New York Times regarding City Island:
Keller Williams Realty Group
There are several communities along the Hudson River without having to cross the Bridge, where I think you might find what you are looking for. Yonkers, Tarrytown, Dobbs Ferry, Hastings and even a little further north, on the Ossining/Briarcliff, Pocantico areas which may give you a little more of the green space that you are looking for.
If you would like to start your search, my contact information is below. Good luck. It is a wonderful time for buyers.
Licensed Real Estate Salesperson
Margot Bennett, Inc.
Licensed Associate Broker
Accredited Buyer Representative
Legends Realty Group